Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Find a value b for the average value of $f(x)=-2x^2+3x+4$ over [0, b] Find a real number b, if it exists, for which the average value of $f(x)=-2x^2+3x+4$ over [0, b] is equal to 5. In other words, on the interval [0,b], there is an average value = 5 for the function $f(x)=-2x^2+3x+4$. I'm mainly checking my answer her... | Since your interval is of the form $[0,b]$, this means that if the average value of $f(x)$ over said interval is $5$, then you get the equation
$$
5 = \frac{1}{b-0}\int_{0}^{b} -2x^2 +3x+4 \ dx
$$
Which, as you pointed out, gives you
\begin{align*}
5 = \frac{1}{b} \left(-\frac{2}{3}x^3 + \frac{3}{2} x^2 + 4x \right)\Bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4047695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The Infinitely Nested Radicals Problem and Ramanujan's wondrous formula In mathematics, a nested-radical is any expression where a radical (or root sign) is nested inside another radical, e.g. $\sqrt{2 + \sqrt{3}}$.
By extension, an infinitely nested radical (aka, a continued root) is an expression where infinitely man... | Method 1:- (More General)
As $$(x+a)^2=x^2+a^2+2ax$$
Replacing $a$ by $n+a$ in above equation where $x,n,a\in R$
$$(x+(n+a))^2=x^2+(a+n)^2+2x(a+n)$$
$\ \implies(x+(n+a))^2=x^2+(a+n)^2+2ax+2nx$
$\ \implies(x+(n+a))^2=ax+(a+n)^2+x^2+2nx+ax$
$\ \implies(x+(n+a))^2=ax+(a+n)^2+x(x+2n+a)$
Taking positive square roots on both... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4049946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
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Finding the Tree for Prufer Code This is more of a fun question that I came up with. How would you go about building the tree for the prufer code:
$P(t) = 122333444455555666666777777788888888999999999(10)(10)(10)(10)(10)(10)(10)(10)(10)(10)$
| I would take the easy way out and use Wolfram Alpha:
ResourceFunction["PruferCodeToLabeledTree"][{1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10}]
Resulting in
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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In simplifying $\sqrt{\frac{(x^2 +x +3)^2}{(1-2q)^2}}$ to $\frac{x^2 +x +3}{|1-2q|}$, why use the absolute value? I have seen in a question
$$\sqrt{\frac{(x^2 +x +3)^2}{(1-2q)^2}}$$ was given to be
$$\frac{x^2 +x +3}{|1-2q|}$$
Why was absolute value given to $1-2q$?
| Well, by definition you have that:
$$
\sqrt{\frac{(x^2 +x +3)^2}{(1-2q)^2}} = \frac{\sqrt{(x^2 +x +3)^2}}{\sqrt{(1-2q)^2}}
$$
and you got that $\forall x \in \mathbb{R}\ \sqrt{x^2} = |x|$
so you get that:
$$
\frac{\sqrt{(x^2 +x +3)^2}}{\sqrt{(1-2q)^2}} = \frac{|(x^2 +x +3)|}{|(1-2q)|}
$$
Clearly $x^2 \geq 0 \ \forall ... | {
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"timestamp": "2023-03-29T00:00:00",
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If $x\geq 0,$ what is the smallest value of the function $f(x)= \frac{4x^2+ 8x + 13}{6(1+ x)}$ If $x\geq 0,$ what is the smallest value of the function
$$f(x)= \frac{4x^2+ 8x + 13}{6(1+ x)}$$
I tried doing it by completing the square in numerator and making it of the form
$$\frac{4(x+ 1)^2+ 9}{6(1+ x)}$$
and then, I pu... | The derivative, $$\frac{2}{3}-\frac{3}{2 (x+1)^2}$$ is zero when $x=1/2$:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4057290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to go about finding whether this limit exists: $\lim_{x \to 0} x [[\frac{1}{x}]]$? I am trying to solve few challenge questions on Real Analysis from Kaczor and Nowak's Problems in Mathematical Analysis, to become more proficient and stimulate thinking. I'd like someone to (a) verify if my proof is correct (b) is t... | I'll make a try to show that $\lim\limits_{x\to0}x[[\frac{1}{x}]]=1$. For $x\not=0$ it is $[[\frac{1}{x}]]\leq \frac{1}{x}<[[\frac{1}{x}]]+1$ hence $\frac{1}{x}-1<[[\frac{1}{x}]]\leq \frac{1}{x}$ which means $1-x<x[[\frac{1}{x}]]\leq 1$ if $x>0$ and $1-x>x[[\frac{1}{x}]]\ge 1$ if $x<0$. So $\lim\limits_{x\to0}x[[\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4062394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove area of Koch snowflake by mathematical induction Here is an interesting construction of a geometric object known as the Koch snowflake.
Define a sequence of polygons $S_0$, $S_1$ recursively, starting with $S_0$ equal to an equilateral triangle with unit sides. We construct $S_{n+1}$ by removing the middle third ... | I want to post the solution, since someone might need it.
Proof. by ordinary induction. Let Induction hypothesis $P(n)$ be $$a_n = a_0\left(\frac{8}{5} - \frac{3}{5}\cdot \left(\frac{4}{9}\right)^n\right).$$
Base case $(n=0):$ $a_0=a_0\left(\frac{8}{5} - \frac{3}{5}\cdot \left(\frac{4}{9}\right)^0\right) = a_0.$ holds.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$ \int _ {- 1} ^ {3} [| x ^ 2-6x | + \text{sgn} (x-2)]\, \text {d} x $
Calculate the integral $$\int _ {- 1} ^ {3} \bigl[| x ^ 2-6x | + \text{sgn} (x-2)\bigr]\, \text {d} x .$$
We know that the two functions are defined as follows,
\begin{align*}
f (x) &= | x ^ 2-6x |\\[5pt]
&=\begin{cases}
x^2-6x, & \text{ if } x^2-... | Let's first work on getting a piecewise equation for the whole thing. There are three points where the pieces can change, namely $x = 0, 2, 6$. So, we will consider each interval separately when writing down the piecewise equation. For example, for $x<0$, we need to consider the following:
$$\begin{cases}
\color{red... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4064373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Big O for error terms The following link from wikipedia explains the Big O notation really good. I have only one problem, which is to formalize the usage of Big O notation for error terms in polynomials. In the example give here we have
$$
e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...=1+x+\frac{x^2}{2!}+\m... | For $0\le x\le1$,
*
*$\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\dfrac{x^5}{5!}+\cdots\le x^3\left(\dfrac1{3!}+\dfrac1{4!}+\dfrac1{5!}+\cdots\right)$ and the sum inside the parenthesis is finite.
*Similarly, $\dfrac{x^2}2+\mathcal O(x^3)\le \dfrac{x^2}2+cx^3\le x^2\left(\dfrac12+c\right).$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given $x^2 +px + q$ has roots -1 & 4, find the values for p & q.
Given $x^2 +px + q$ has roots $-1$ & $4$, find the values for $p$ & $q$.
Attempt:
$$
x = \frac{-p\pm\sqrt{p^2-4q}}{2}\\(p + 2x)^2 = p^2 - 4q\\
p^2 + 4px + 4x^2 = p^2 -4q
\\
q = -x^2 - px
\\
q = -(-1)^2 -p(-1)
\\
q = -1 - p
\\
q = -(4)^2-p(4)
\\
q = -16 ... | From $x^2 + px + q = a(x+1)(x-4) = x^2 -3x -4$ and from comparison of the coefficients follows $a = 1$, $p = -3$ and $q = -4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4066478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to evaluate $ \frac{1}{3} + \frac{2}{3^{2} } + \frac{1}{3^{3}} + \frac{2}{3^{4}} + \frac{1}{3^{5}} + ...$? I was given the series:
$$ \frac{1}{3} + \frac{2}{3^{2} } + \frac{1}{3^{3}} + \frac{2}{3^{4}} + \frac{1}{3^{5}} + ...$$
Making some observations I realized that the $ a_{n} $ term would be the following:
$$ a_... | In base $3$, we can say
$$N = \frac {1}{10} + \frac{2}{100} + \frac{1}{1000} + \frac{2}{10000} + \cdots
=0.121212\cdots$$
Staying in base 3, we get
\begin{array}{rcr}
N &= &0.121212\cdots \\
100N &= &12.121212 \cdots \\
-N &= &-0.121212\cdots \\
\hline
22N &= &12.000000\cdots \\
\end{array}
So $N = \dfrac{12}{22}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4067389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 4
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Computing $\int_0^\infty \frac{\ln x}{(x^2+1)^2}dx$ I'm trying to compute
$$I=\int_0^\infty \frac{\ln x}{(x^2+1)^2}dx$$
The following is my effort,
$$I(a)=\int_0^\infty\frac{\ln x}{x^2+a^2}dx$$
Let $x=a^2/y$ so that $dx=-(a^2/y^2)dy$ which leads to
$$I(a)=\int_0^\infty \frac{\ln(a/y)}{a^2+y^2}dy=\int_0^\infty\frac{\ln... | Here is an easy way to compute it :
Let's substitute $ \left\lbrace\begin{matrix}y=\frac{1}{x}\ \ \\ \mathrm{d}x=-\frac{\mathrm{d}y}{y^{2}}\end{matrix}\right. $, we get : \begin{aligned}\int_{0}^{+\infty}{\frac{\ln{x}}{\left(x^{2}+1\right)^{2}}\,\mathrm{d}x}&=-\int_{0}^{+\infty}{\frac{y^{2}\ln{y}}{\left(1+y^{2}\right)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4069094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 3
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Find the value of $T=\mathop {\lim }\limits_{n \to \infty } {\left( {1+ \frac{{1+\frac{1}{2}+ \frac{1}{3}+ . +\frac{1}{n}}}{{{n^2}}}} \right)^n}$ I am trying to evaluate
$$T = \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{{1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}}}{{{n^2}}}} \right)^n}.$$
My sol... | Your solution is incorrect. The good solution is that $\sum_{k=1}^n \frac1k \sim \log n,$ so that
$$ 1\leq T = \mathop {\lim }\limits_{n \to \infty } {\left( {1 + \frac{{1 + \frac{1}{2} + \frac{1}{3} + . + \frac{1}{n}}}{{{n^2}}}} \right)^n} \leq \mathop {\lim }\limits_{n \to \infty }\left(1 + \frac{\log n}{n^2} \right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4071409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 4
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What will happen to $\theta$ when $\sin \theta =\frac{\sqrt {{2{\pm}\sqrt i}}}{2}$ and i >4? Some follow up questions related to Two mysterious missing angles in the sine values of acute angle list?
.After finding two missing angles, we have a list of 9 sine values for special angles between 0$^\circ$ to $90^\circ$:... | About your first question about why they are symmetric about $45^\circ$. Let's say that you have an angle that is $x^\circ$ from the $45^\circ$ angle to make an angle of $45-x^\circ$. Thus, the symmetric angle must be $45+x^\circ$. So, if you add these angles up, you get $90^\circ$, which means that they are complement... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4072192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Why do we multiply by $-1$ here? WolframAlpha solves $$\sin\left(x-\fracπ4\right)=\frac{1+\sqrt3}{2\sqrt2}$$ by multiplying by $-1$ as such:
$$\sin\left(-x+\fracπ4\right)=-\frac{1+\sqrt3}{2\sqrt2}$$ then arcsin both sides, etc. but why multiplying by $-1$ and not just directly find the answer? Mainly say this because I... | $\sin(x)$ is an odd function, so $\sin(x) = -\sin(-x)$. This means that
\begin{align*}
\sin\left(x-\frac{\pi}{4}\right)=\frac{1+\sqrt3}{2\sqrt2} &\iff -\sin\left(-x+\frac{\pi}{4}\right)=\frac{1+\sqrt3}{2\sqrt2}\\
&\iff\sin\left(-x+\frac{\pi}{4}\right)=-\frac{1+\sqrt3}{2\sqrt2}.
\end{align*}
In other words, both equati... | {
"language": "en",
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Separating scaling, rotation and sheer coefficients correctly in this case This question follows-on from Correct the Fourier transform of a shear-distorted image? ($x_{new} = x + \alpha y$). I have a group of distorted $x, y$ points and have found a matrix $A$ that corrects their positions:
$$
\begin{bmatrix}
x... | You have already observed the bad effect if you use $-s$ and $s$ in the off-diagonal terms of your "shear" matrix (making an antisymmetric matrix) -- the matrix doesn't just look like a rotation matrix, it actually is a combined rotation and scaling.
Changing the form of the matrix to a symmetric matrix does not help, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4081235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Applying Gauss' Divergence Theorem to given integral We are given the following two integrals:
$$\iint\limits_S D_n f\:dS $$ and
$$\iiint\limits_B \nabla \cdot (\nabla f) \: dV$$
where $S$ is the portion of the sphere $x^2+y^2+z^2=a^2$ in the first octant, $n$ is the unit normal vector to $S$ at $(x,y,z)$ and $f(x,y,z)... | To apply divergence theorem, you must have a closed surface. So we close the surface by placing $3$ quarter disks in plane $x = 0, y = 0, z = 0$.
Please note that when you are doing volume integral, you cannot equate $\nabla \cdot (\nabla f) = \displaystyle \frac{2}{x^2+y^2+z^2} = \frac{2}{a^2}$. It should rather be,
$... | {
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Find all natural solutions that satisfy $2^ + 3^ = ^2$ It looks like an easy question but I couldn't find a way to solve it. I found (0,1,2),(3,0,3),(4,2,5) by trial and error and I'm kinda sure they are the only answers but I'm not sure how to prove it.
| First, let us solve the equation $3^a = 2^b-1$, where $a, b \in \mathbb{N}$.
For $b = 0$ we have no solutions.
For $b = 1$ we obtain $a = 0$. For $b = 2$ we get $a=1$. If $b \ge 3$ then $2^b-1 \equiv -1(\text{mod } 8)$ and hence we have no other solutions, since $3^a \equiv 1,3(\text{mod } 8)$.
In conclusion, we get th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4082057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Solving $(\sqrt{2})^x+(\sqrt{2})^{x-1}=2(2\sqrt{2}+1)$ I'm in stuck with this simple equation.
$$(\sqrt{2})^x+(\sqrt{2})^{x-1}=2(2\sqrt{2}+1)$$
This is my solution:
$$\begin{align}(\sqrt{2})^x+(\sqrt{2})^x(\sqrt{2})^{-1} &=4\sqrt{2}+2 \tag{1}\\[4pt]
2^{x/2}+2^{(x-1)/2}&=2^2\cdot 2^{1/2}+2^1 \tag{2}\\[4pt]
{\frac x2}+{\... | Your incorrect step starts from here.
$$2^{\frac x2}+2^{\frac{x-1}{2}}≠\frac x2+\frac{x-1}{2}$$
Because, this is not a equality:
$$2^a+2^b ≠ a+b$$
You can do one of the correct solutions as follows:
$$(\sqrt 2)^{x-1}(\sqrt 2-1)=4\sqrt 2+2$$
$$(\sqrt 2)^{x-1}=\frac{4\sqrt 2+2}{\sqrt 2-1}$$
$$(\sqrt 2)^{x-1}=\frac{(4\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4083390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Maximum possible value with 3 variables and fractional equations A cool problem I was trying to solve today but I got stuck on:
Find the maximum possible value of $x + y + z$ in the following system of equations:
$$\begin{align}
x^2 – (y– z)x – yz &= 0 \tag1 \\[4pt]
y^2 – \left(\frac8{z^2}– x\right)y – \frac{8x}{z^2}&=... | From (1):
\begin{align*}
0 &= x^2 -(y-z)x-yz \\
&= (x+y+z)(x-y) -y(x-y) \text{,}
\end{align*}
so either
$$ x+y+z = y \qquad \text{ or } \qquad x-y = 0 \text{.} $$
We conclude either $x = -z$ or $x = y$.
From (2):
\begin{align*}
0 &= z^2 y^2 - (8-xz^2)y-8x \\
&= (x+y+z)(yz^2 - 8) - z(yz^2-8) \text{,} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4084624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Prove whether the following series is convergent or divergent Prove the convergence of the following series:
$$\sum\frac{1 + 2^n + 5^n}{3^n}$$
My idea was to write it as:
$$\frac{1}{3^n} + \frac{2^n}{3^n} + \frac{5^n}{3^n}$$
Which would mean that $\frac{1}{3^n}$ is a geometric series with a ratio of $\frac{1}{3}$, and ... | Simply write $\frac{5^n}{3^n}$ as $\left(\frac53\right)^n$. Then,
$$\sum \frac{1+2^n+5^n}{3^n}>\sum \frac{5^n}{3^n}=\sum\left(\frac53\right)^n>\sum 1^n=\infty$$
So the entire sum diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4085711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I prove this formula $\int_0^\infty f\left( a^2x^2+\frac{b^2}{x^2}\right)\mathrm{d}x=\frac{1}{a}\int_0^\infty f(x^2+2ab)\mathrm{d}x$? While solving this trigonometric integral :
$$\int_0^\infty \sin\left(a^2x^2+\frac{b^2}{x^2}\right)\mathrm{d}x$$
I came across this formula :
$$\int_0^\infty f\left( a^2x^2+\frac... | Assume that $a,b > 0$. If $\displaystyle{t = a x - \frac{b}{x}}$, then
$$t^2 + 2 a b = a^2 x^2 + \frac{b^2}{x^2} - 2 ab + 2 a b =
a^2 x^2 + \frac{b^2}{x^2}.$$
Also one has
$$dt = \left(a + \frac{b}{x^2} \right) dx$$
as $x$ varies from $0$ to $\infty$, we see that $t$ varies from $-\infty$ to $\infty$. Hence, as long a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4085987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Solution verification about complex multiplication in polar form I am trying to prove the process behind complex multiplication in polar form, which is similar to this:
The product of two complex numbers in polar form $r_{1}\,\angle\,\theta_{1}$ and $r_{2}\,\angle\,\theta_{2}$ is $r_{1}r_{2}\,\angle\,(\theta_{1} + \th... | Your computations are correct, but you do not cover all possible cases. Problems are
*
*Your formulae based on $\tan^{-1} = \arctan$ only work if both $a \ne 0$ and $c \ne 0$. This reflects the fact $\tan x$ is not defined if $x = k \pi/2$ with odd $k \in \mathbb Z$. Thus you must separately treat a number of special... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4087731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Taylor series for $e^{-x \ln x}$ What is a Taylor expansion for the following function?
$$ e^{-x \ln x} $$
I assume you can't do a Taylor expansion around $x=0$, since the function doesn't exist at that point. The next best choice seems to be $x=1$, since that's when $\ln(x)=0$. So, when I try to do an expansion arou... | We have
\begin{align*}
\color{blue}{x^{-x}}&=1 - (x - 1) + \frac{1}{2} (x - 1)^3 - \frac{1}{3} (x - 1)^4 + \frac{1}{12}(x - 1)^5 + \frac{1}{120}(x - 1)^6 - \dots\\
&=\sum_{n=0}^\infty a_n\frac{(x-1)^n}{n!}
\end{align*}
when expanded at $x=1$,
where
\begin{align*}
\color{blue}{(a_n)_{n\geq 0}=(1, -1, 0, 3, -8, 10, 6,... | {
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} |
Does the system of two equation in two variables given below have a possible real solution? Consider the following two equations:
$$\sqrt {3x} \left(1 + \frac{1}{x+y} \right) = 2 $$
$$\sqrt{7y}\left(1 -\frac{1}{x+y}\right)= 4\sqrt{2}$$
.
It is expected to verify if the equations possess a solution and if yes what is th... | Starting from @Moo's comment $$1323 x^4+3024 x^3+2610 x^2-7456 x+147 = 0$$ is the same as
$$\left(9 x^2+30 x+49\right) \left(147 x^2-154 x+3\right)=0$$ The first quadratic does not show real roots and for the second
$$x_\pm=\frac{11\pm4 \sqrt{7}}{21} $$
Similarly
$$3087 y^4-25284 y^3+44716 y^2-38816 y+4032 = 0$$is the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4090737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Approximate result for $\prod_{n=1}^\infty\left(1-\frac{1}{2^n}\right)$? What would be a quick way to approximately determine the value of
$$\prod_{n=1}^\infty\left(1-\frac{1}{2^n}\right)=\phi\left(\frac{1}{2}\right) , $$
where $\phi(q)$ is the Euler function? By approximating, I mean determine the first few digits of ... | $$\prod_{n=1}^\infty\left(1-\frac{1}{2^n}\right)=\Bigg[\prod_{n=1}^p\left(1-\frac{1}{2^n}\right)\Bigg]\Bigg[\prod_{n=p+1}^\infty\left(1-\frac{1}{2^n}\right)\Bigg]$$
Fot the second product, take its logarithm
$$\log\Bigg[\prod_{n=p+1}^\infty\left(1-\frac{1}{2^n}\right)\Bigg]=-\sum_{n=p+1}^\infty \left(\frac{1}{x}+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4093369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove $\{a_n\}$ converges.
Suppose $a_1,a_2>0$ and
$a_{n+2}=2+\dfrac{1}{a_{n+1}^2}+\dfrac{1}{a_n^2}(n\ge 1)$. Prove
$\{a_n\}$ converges.
First, we may show $\{a_n\}$ is bounded for $n\ge 3$, since $$2 \le a_{n+2}\le 2+\frac{1}{2^2}+\frac{1}{2^2}=\frac{5}{2},~~~~~~ \forall n \ge 1.$$
But how to go on?
| I'm not so sure about this one, but here's my attempt-
\begin{align}
\lVert a_{n+1}-a_n \rVert &= \left\lVert \frac{1}{a_n^2}-\frac{1}{a_{n-2}^2} \right\rVert\\
&\leq \frac{ \lVert {a_{n-2}^2}-a_n^2 \rVert }{16} \\
&= \frac{ \lVert {a_{n-2}+a_{n}} \rVert \lVert {a_{n-2}-a_{n}} \rVert }{16} \\
&\leq \frac{6}{16} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4094333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
irrational integral $ \int \frac{1+\sqrt{x^2+3x}}{2-\sqrt{x^2+3x}}\, dx$ I have to solve this irrational integral $$ \int \frac{1+\sqrt{x^2+3x}}{2-\sqrt{x^2+3x}}\, dx$$
It seems that the most convenient way to operate is doing the substitution
$$ x= \frac{t^2}{3-2t}$$
according to the rule,
obtaining the integral:
$$ \... | In your integral ,you can make break the numnerator in terms of denominator by first putting a -ve sign in front of the integral , then proceed .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4094912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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If x, y, w, z >0 and $x^4$+$y^4$+$w^4$+$z^4$ <=4 prove 1/$x^4$+1/$y^4$+1/$w^4$+1/$z^4$>=4 I would appreciate suggestions to solve:
If x, y, w, z > 0 and $x^4$ + $y^4$ + $w^4$ + $z^4$ <=4 prove the following:
1/$x^4$ + 1/$y^4$ + 1/$w^4$ + 1/$z^4$ >= 4
From plugging in numbers into Excel, it looks like x, y, w, z must be... | Thank you very much for the hints. Here is the solution from the hint of achille hui:
We are told that $x^4$ + $y^4$ + $w^4$ + $z^4$ <=4 so ($x^4$+$y^4$ + $w^4$ + $z^4$)/4 <= 1
But AM >= HM (or HM <= AM) so:
4/(1/$x^4$ + 1/$y^4$ + 1/$w^4$ + 1/$z^4$) <= ($x^4$+$y^4$ + $w^4$ + $z^4$)/4 <= 1
Which means:
4/(1/$x^4$ + 1/$y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4096004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How can I solve $x\left(y^2+z\right)z_x-y\left(x^2+z\right)z_y=\left(x^2-y^2\right)z$? I have this right now:
$$x\left(y^2+z\right)z_x-y\left(x^2+z\right)z_y=\left(x^2-y^2\right)z$$
$$\frac{dx}{x\left(y^2+z\right)}=\frac{dy}{-y\left(x^2+z\right)}=\frac{dz}{\left(x^2-y^2\right)z}$$
I get the first first integral like th... | Take the first ODE:
$$\frac{dx}{x\left(y^2+z\right)}=\frac{dy}{-y\left(x^2+z\right)}$$
$$y(x^2+z)dx+x(y^2+z)dy=0$$
$$\dfrac 12xy(2xdx+2ydy)+z(ydx+xdy)=0$$
$$\dfrac 12xy(dx^2+dy^2)+z(ydx+xdy)=0$$
$$\dfrac 12xyd(x^2+y^2)+zd(xy)=0$$
And eliminate the $z$ variable since you have:
$$x^2+y^2-2z=C_1$$
$$z=\dfrac {x^2+y^2-C_1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4097001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Integrate $\int\frac{3x^2-1}{\sqrt{x^2+x-1}}dx$ Integrate $\int\frac{3x^2-1}{\sqrt{x^2+x-1}}dx$
I solved this integral by euler substitution by replacing
$\sqrt{x^2+x-1}=x+t$
but it's not allowed by the problem.
p.s Is there any other method to solve with?
Thank you in advance :)
| $$
I = \int\frac{3x^2-1}{\sqrt{x^2+x-1}}dx = \int\frac{3x^2-1}{\sqrt{(x + \frac12)^2 - \frac54}}dx
$$
Replacing $x= y\cdot \sqrt{\frac54} - \frac12$ gives
$$
I = \int\frac{(15 y^2)/4 - (3 \sqrt 5 y)/2 - 1/4}{\sqrt{(y^2-1)}}dy
$$
Now let $y = \cosh (z)$ which gives
$$
I = \int [15 (\cosh (z))^2)/4 - (3 \sqrt 5 \cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4100874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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$2+\log _{2} a=3+\log _{3} b=\log _{6}(a+b)$ The problem
Given that $a,b>0$ and $$2+\log _{2} a=3+\log _{3} b=\log _{6}(a+b)$$
Find the value of $$\log _{a b}\left(\frac{1}{a}+\frac{1}{b}\right)$$
My attempt
We have from the given condition
$$2+\frac{\log a}{\log 2}=3+\frac{\log b}{\log 3}=\frac{\log (a+b)}{\log 6}... | Set
\begin{align}\\ \frac{2\log 2+\log a}{\log 2}=\frac{3 \log 3+\log b}{\log 3}=\frac{\log (a+b)}{\log 6}=y\\\to \log\left(\frac{1}{a}+\frac{1}{b}\right)=\log(108) \to 108ab=a+b\end{align}
This is where you are.
Let us find another equation to eliminate a or b.
From \begin{align}\\2\log 2\log 3+\log a\log 3=3 \log ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4103740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Calculating a limit. Is WolframAlpha wrong or am I wrong? What I'm trying to solve:
$$\lim _{x\to -\infty \:}\frac{\left(\sqrt{\left(x^2+14\right)}+x\right)}{\left(\sqrt{\left(x^2-2\right)}+x\right)}$$
What I put into WolframAlpha:
(sqrt(x^2+14)+x)/(sqrt(x^2-2)+x)
My result: $1$, which I get by simply dividing bot the ... | You can divide by $x$ but not in that way. Since $x\to-\infty$ it is eventually negative i.e. $-x>0$ so you should have $$\frac{\sqrt{x^2+c}}x = \frac{\sqrt{x^2+c}}{-\sqrt{x^2}}= -\sqrt{1+\frac c{x^2}}=-1-\frac c{2x^2}+O(|x|^{-3})$$
This last inequality via the approximation $\sqrt{1+t}= 1 + \frac t2+O(t^2)$ which can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4105217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
RMS value of trapeziodal fourier series I calculated RMS value of trapezoidal fourier series but the numerical results are not same with its normal formula.
$b_n = \frac{8\cdot A}{\pi \cdot u \cdot n^2}\cdot sin(\frac{n \cdot u}{ 2})$
$f_{rms} = \sqrt{ a_0^2 + \frac{a_1^2 + a_2^2 +a_3^2+..... + b_1^2 + b_2^2 + b_3^2}{2... | Using Octave code similar to those in Shifted square wave Fourier series
format long;
A = 12;
u = pi / 6;
t = 0:0.0001:(2*pi);
n = (1:2:1000).';
fourier_bn = @(n) 8 .* A ./ (pi .* u .* n.^2) .* sin(n .* u ./ 2);
fourier_term = @(n) fourier_bn(n) .* sin(n .* t);
fourier = sum(cell2mat(arrayfun(fourier_term, n, 'Unifo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4106722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Integral $\int_0^\infty \frac{x^{2k}}{x^2+1}dx$ I am little bit confused by the following integral:
$$\int_0^\infty \frac{x^{2k}}{x^2+1}dx,$$
which according to WA is equal to
$$\int_0^\infty \frac{x^{2k}}{x^2+1}dx=\frac{\pi}{2}\sec(\pi k),\quad \text{for}\ \operatorname{Re}(k)>-\frac{1}{2}.$$
However, by plugging $k=1... | Letting $\displaystyle \frac{1}{t}=x^{m}+1$, then $
\displaystyle d x=\frac{1}{m} \left(\frac{1}{t}-1\right)^{\frac{1}{m}-1}\left(-\frac{1}{t^{2}}\right) d t.
$
Consequently
$$
\begin{aligned}
\int_{0}^{\infty} \frac{x^{r}}{x^{m}+1} d x &=-\int_{1}^{0} \frac{\left(\frac{1}{t}-1\right)^{\frac{r}{m}}}{\frac{1}{t}} \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4107579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Evaluate $\int \frac{x^3+4x^2+x-1}{x^3+x^2}dx$ Evaluate $\int \frac{x^3+4x^2+x-1}{x^3+x^2}dx$.
Where do I start with this integral? I can easily see that it is possible to fator $x^{2}$ out on the denominator and use partial fractions. The numerator is also factorable but it does not have any integer roots. Can someone... | Simplify step by step:
$$\begin{align}
\int \frac{x^3+4x^2+x-1}{x^3+x^2}dx
&= \int 1+ \frac{3x^2+x-1}{x^3+x^2}dx \\
&= x + \int\left(\frac{3x^2 + 2x}{x^3+x^2} -\frac{x + 1}{x^3+x^2}\right)dx \\
&= x + \int\frac{3x^2 + 2x}{x^3+x^2}dx - \int\frac{x + 1}{x^3+x^2}dx \\
&= x + \ln|x^3+x^2| - \int\frac{1}{x^2}dx \\
&= x+\ln... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4110066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find minimum of ${(a+3c)\over(a+2b+c)} + {4b\over(a+b+2c)} - {8c\over(a+b+3c)}$ for non-negative reals
Let $a, b, c \ge 0$, not all zero. Find the minimum value of $${(a+3c)\over(a+2b+c)} + {4b\over(a+b+2c)} - {8c\over(a+b+3c)}.$$
Here was my attempt:
If c tends to infinity and a and b are small we get 1/3.
Now , i t... | As Albus mentioned, we use the substitution:
$ x = a + 2b + c, y = a + b + 2c, z = a + b + 3c$.
This system gives us $ a = -x + 5y - 3z, b = x - 2y + z, c = 0x - y + z$.
The expression becomes:
$$\frac{ -x + 2y } { x} + \frac{ 4 x - 8 y + 4z } { y} + \frac{ 8y - 8z } { z} = \frac{ 2y}{x} + \frac{ 4x}{y} + \frac{4z}{y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4110847",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$ \int \frac{1}{2+\sqrt{x+1}+\sqrt{3-x}} \cdot d x $ I am trying to evaluate this antiderivative $$
\int \frac{1}{2+\sqrt{x+1}+\sqrt{3-x}} \cdot d x
$$
What i have done:
$$
\begin{split}
I
&= \int \frac{1}{2+\sqrt{x+1}+\sqrt{3-x}} \cdot d x \\
&= \int \frac{2+\sqrt{x+1}-\sqrt{3-x}}{4+x+1+4 \sqrt{x+1}-3+x} \cdot d x\\... | Solution without trigonometry:
Let $u=\sqrt{x+1}$ and $v=\sqrt{3-x}$, then $u^2+v^2=4$ and $u^2-v^2=2x-2$, so $udu+vdv=0$ and $dx=udu-vdv=2udu=-2vdv$.
\begin{align*}\int\frac{dx}{2+u+v}&=\int\frac{(u+v-2)dx}{(u+v)^2-4}=
\int\frac{(u+v-2)dx}{2uv}=\int\left(\frac{dx}{2v}+\frac{dx}{2u}-\frac{dx}{uv}\right)\\&=\int\left(-d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4111706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Solve the equation $\frac{x-13}{x-14}-\frac{x-15}{x-16}=-\frac{1}{12}$ Solve the equation $$\dfrac{x-13}{x-14}-\dfrac{x-15}{x-16}=-\dfrac{1}{12}.$$
For $x\ne14$ and $x\ne 16$ by multiplying the whole equation by $$12(x-14)(x-16)$$ we get: $$12(x-16)(x-13)-12(x-14)(x-15)=-(x-14)(x-16).$$ This doesn't look very nice. Can... | Hint: Note that
$$\frac{x-13}{x-14} = \frac{x-14+1}{x-14} = 1 + \frac{1}{x-14} \tag{1}\label{eq1A}$$
$$\frac{x-15}{x-16} = \frac{x-16+1}{x-16} = 1 + \frac{1}{x-16} \tag{2}\label{eq2A}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4117579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Solve the equation $\frac{x^2-10x+15}{x^2-6x+15}=\frac{4x}{x^2-12x+15}$ Solve the equation $$\dfrac{x^2-10x+15}{x^2-6x+15}=\dfrac{4x}{x^2-12x+15}.$$
First we have $$x^2-6x+15\ne0$$ which is true for every $x$ ($D_1=k^2-ac=9-15<0$) and $$x^2-12x+15\ne0\Rightarrow x\ne6\pm\sqrt{21}.$$ Now $$(x^2-10x+15)(x^2-12x+15)=4x(x^... | Componendo and dividendo yields:
$$\frac{x^2-6x+15}{x^2-10x+15} = \frac{x^2-12x+15}{4x}$$
$$\Rightarrow \frac{4x}{x^2 - 10x + 15} = \frac{x^2-16x+15}{4x} \tag{$\frac{a-b}{b} = \frac{c-d}{d}$}$$
$$\Rightarrow u=x^2-13x+15: \frac{4x}{u + 3x} = \frac{u - 3x}{4x}$$
$$\Rightarrow u^2 - 9x^2 = 16x^2$$
$$\Rightarrow (u - 5x)(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4119790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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How the right term of the derivative is gained? This deduction is one of the typical ones I think.
What I want to deduce is the right term from the left term of the below equation.
$$\frac{d}{dx}\left(\log\left(\frac{a+\sqrt{a^{2}+x^{2}}}{x}\right)\right)=\frac{-a}{x\sqrt{a^{2}+x^{2}}}$$
$\frac{d}{dx}\left(\log\left(\f... | Small error: a missing + sign in the fourth line from the product rule:
$$=\left(\frac{x}{a+\sqrt{a^{2}+x^{2}}}\right)\left((x^{-1})'\left(a+\sqrt{a^{2}+x^{2}}\right) ~~\mathbf{+}~~ x^{-1}\frac{d}{dx}\left(a+\sqrt{a^{2}+x^{2}}\right)\right)$$
Continuing on with your idea, the next step would be to use the chain rule on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4120409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Pythagorean triples conditions Pythagorian triple is every triple of natural numbers $(x, y, z)$ such that $x, y, z$ are sides of a right triangle, where $z$ is the hypotenuse.
Now, Pythagorean theorem says:
$$x^2 + y^2 = z^2 \tag1$$
If we look just natural solutions to the equation $(1)$, without geometrical condition... | One other way to analyse the equation $x^2+y^2=z^2$ is to solve it in its quadratic form.
Assuming $x\ne y$ otherwise $z$ would be irrational with factor $\sqrt{2}$.
Lets take $(x<y<z)>0$ without loss of generality and define $x=y-m$ and $z=y+n$ where $m$ and $n$ are positive integers. Giving
$$(y-m)^2+y^2=(y+n)^2\tag{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4121977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Study the series of $\int_0^{\frac{1}{n^a}}\sin{(\sqrt[3]{x})}\,dx$ with respect to $a>0$ I have to study the series $\sum a_n$ with $a_n=\int_0^{\frac{1}{n^a}}\sin{(\sqrt[3]{x})}\,dx$, with respect to $a>0$.
I have thought to use the asymptotic criterion for series.
In particular I can observe that:
$\sin{x}\sim x-\fr... | Here is a more rigourous version of what you did. Using the Lagrange form of the remainder term in Taylor's formula, we have
$$
\left| {\sin w - w} \right| \le \frac{{w^3 }}{6}
$$
for all $w\geq 0$. Consequently,
\begin{align*}
\left| {a_n - \frac{3}{{4n^{4a/3} }}} \right| &= \left| {\int_0^{1/n^a } {(\sin \sqrt[3]{x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4122996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Solve the following system of equations:$|x|+|y|=2$ and $y=x+1$.
Solve the following system of equations:
$|x|+|y|=2$ and $y=x+1$, where $x$ is a real number.
Approach:
I substituted $y$ in equation $1$, so:
$$ \ \ \ \ \ \ \ \ \ \ |x|+|x+1|=2$$
$$1 \ \ \ \ \ \ \ \ \ \ 1$$
$$2 \ \ \ \ \ \ \ \ \ \ 0$$
$$0 \ \ \ \ \ \ \... | HINT
*
*If $x≥0$, then $y=x+1>0$
$$\begin{cases} x+y=2 \\ y=x+1 \end{cases}$$
Then, we need
*
*If $-1≤x<0, ~y≥0 $
$$\begin{cases} y-x=2 \\ y=x+1 \end{cases}$$
*
*If $x<-1, ~y<0$
$$\begin{cases} -x-y=2 \\ y=x+1 \end{cases}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4124031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Determine the limit using Taylor expansions: I struggle with this one—maybe someone could point me in the right direction.
$$\lim_{x\to 0} \frac{5^{(1+\tan^2x)} -5}{1-\cos^2x}$$
Getting the Taylor series expansion for $\tan^2x$ and $\sin^2x$ is no problem, but I struggle with getting further along at this step:
$$\frac... | $\lim\limits_{x\to0}\dfrac{5^{1+\tan^2(x)}-5}{1-\cos^2(x)}=$
$=5\lim\limits_{x\to0}\dfrac{5^{\tan^2(x)}-1}{\sin^2(x)}=$
$=5\lim\limits_{x\to0}\left[\dfrac{5^{\tan^2(x)}-1}{\tan^2(x)}\cdot\dfrac1{\cos^2(x)}\right]=$
$=5\lim\limits_{x\to0}\dfrac{5^{\tan^2(x)}-1}{\tan^2(x)}\cdot\lim\limits_{x\to0}\dfrac1{\cos^2(x)}\unders... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4124686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Proving $\sin P+\sin Q=2\sin\frac{P+Q}2\cos\frac{P-Q}2$ using the sin addition formula and double angle formula This question asks me to prove the validity of the trig identity by using the compound angle identities for $\sin(A+B)$ and $\sin(A-B)$
$$\sin P +\sin Q =2\sin\frac{P+Q}{2}\cos\frac{P-Q}{2}$$
I didn't get it ... | They did it because they could. $A$ and $B$ don't have any significance on their own. If you have any two numbers $x,y$ and can always find two other numbers where $x= A+B$ and $y= A-B$. (just let $A=\frac {x+y}2$ and $B=\frac {x-y}2$). This can be a useful manipultion to make as expression appear simpler.
But we co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4131951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$\sum\left(\frac{ n^2 + 1}{n^2 +n + 1}\right)^{n^2} $ converges or diverges? The original question is to show that $\;\sum\left(\dfrac{ n^2 + 1}{n^2 +n + 1}\right)^{n^2} $ either converges or diverges.
I know it diverges but I'm having difficulty arriving at something useful for $ a_n $.
Here's what I did:
$ a_n = \l... | Let's use the root test:
$$\sqrt[\Large n]{\left( \frac{n^2+1}{n^2+n+1} \right)^{n^2}} = \left( \frac{n^2+1}{n^2+n+1} \right)^{n} = \left[\left( 1-\frac{n}{n^2+n+1} \right)^{-\frac{n^2+n+1}{n}}\right]^{\frac{-n^2}{n^2+n+1}}\stackrel{n\to\infty}{\longrightarrow} e^{-1} < 1,$$
and therefore the series converges.
We used:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4132689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
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For what values of $a$ is the vector $(a ^ 2, a, 1)$ in $\langle (1,2,3), (1,1,1), (0,1,2) \rangle$? For what values of $a$ is the vector $(a ^ 2, a, 1)$ in $\langle (1,2,3), (1,1,1), (0,1,2) \rangle$?
My question is, should I find scalars $ \alpha, \beta, \gamma $ such that $ \alpha (1,2,3) + \beta (1,1,1) + \gamma (0... | $ \alpha (1,2,3) + \beta (1,1,1) + \gamma (0,1,2) = (a ^ 2, a, 1) $
this becomes the following system of equations:
$\alpha + \beta +0 \gamma=a^2$
$2\alpha + \beta + \gamma=a$
$3\alpha + \beta +2 \gamma=1$
Solving by gauss elimination:
First write the system in augmented matrix form
$
\begin{bmatrix}
1 & 1 & 0 & a^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4139233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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How to solve this binomial summation Problem where I was stuck at:
$$\sum_{k=0}^{n} (-4)^k\cdot \binom{n+k}{ 2k },$$ what I tried was to get observe that inside term is coefficients in expansion of $(1+2x)^{n+k}$ but evaluating it afterwards is not working for me.
| We seek to evaluate
$$\sum_{k=0}^n (-1)^k 2^{2k} {n+k\choose 2k}$$
which is
$$\sum_{k=0}^n (-1)^k 2^{2k} {n+k\choose n-k}
= [z^n] (1+z)^n \sum_{k=0}^n (-1)^k 2^{2k} (1+z)^k z^k.$$
Here the coefficient extractor enforces the upper limit of the sum and
we find
$$[z^n] (1+z)^n \frac{1}{1+4z(1+z)} =
[z^n] (1+z)^n \frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4143496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Given $x+y+z=1,x^2+y^2+z^2=2,x^3+y^3+z^3=3,$ can we conclude that $x^n+y^n+z^n\in\mathbb{Q}$ for all $n\in\mathbb{N}$. Is there an explicit form? I had seen a fun problem that is exactly the problem in the question, except it was a specific case of this. Turns out, if $x+y+z=1,x^2+y^2+z^2=2,x^3+y^3+z^3=3,$ then $x^5+y^... | Yes. In https://en.m.wikipedia.org/wiki/Symmetric_polynomial, look at Power-sum Symmetric Polynomials in the section Special kinds of symmetric polynomials
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4145263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Algebra trick clarification question I have the equation $x = \sqrt{3} + \sqrt{2} \,\,$ and the trick is that $\frac{1}{x} = \sqrt{3} - \sqrt{2}$. I don't see how the minus sign appears just by taking 1 over x?
| If we are told that $x = \sqrt{3}+\sqrt{2}$ then it follows that $\dfrac{1}{x} = \dfrac{1}{\sqrt{3}+\sqrt{2}}$
$=\dfrac{1}{\sqrt{3}+\sqrt{2}}\cdot\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} = \dfrac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}=\dfrac{\sqrt{3}-\sqrt{2}}{(\sqrt{3})^2-(\sqrt{2})^2} = \dfrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4145617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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How many triangles are there whose sides are prime numbers and have a perimeter equal to $29$?
Perimeter of a triangle is equal to $29$ and the values of all three sides are prime number. How many many incongruent triangles we have with this properties?
$1)2\qquad\qquad\qquad2)3\qquad\qquad\qquad3)4\qquad\qquad\qqua... | Modulo $6$ the residues of the primes are $3,-1,1,-1,1$ respectively – and our target has residue $-1$ modulo $6$. If we use two $3$s the last side must be $23$, which is impossible. If we use one $3$ the other two sides must have residue $1$ each and sum to $26$ – the only possibility is $13$ and $13$. If no $3$s, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4150737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find $x$ for $3^x = 5/3$ Why my answer is wrong? $3^x = 5/3$
$e^{\ln 3^x} = 5/3$
$e^{x \ln 3}=5/3$
$e^x \times e^{\ln 3} = 5/3$
$e^x \times 3 = 5/3$
$e^x = 5/9$
$x = \ln(5/9)$
| We know that $a^x = e^{x \ln a}$
thus $3^x = e^{x \ln 3} = 5/3$
applying $\ln$ to both sides, we get $x \ln 3 = \ln (5/3) = \ln(5) - \ln(3)$
The rest is straightforward algebra:
$$x \ln 3 + \ln 3 - \ln 5 = 0$$
$$(x+1) \ln 3 = \ln 5$$
$$x + 1 = \ln 5 / \ln 3$$
$$x = \frac{\ln 5}{\ln 3} - 1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4154947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Paradox of $-i$ seemingly equal to $1$ via the Wallis product for $\pi$ and the Euler sine product Assuming $x$ is a real variable throughout,$$\frac{\sinh(ix)}{i}=\sin(x)$$
$$\frac{\sinh(\pi ix)}{\pi ix} = \frac{\sin(\pi x)}{\pi x}$$
$$\frac{e^{\pi ix}-e^{-\pi ix}}{2\pi ix}=\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\... | You appear to be making a mistake in the left hand side of your fourth equation. $e^{\frac{1}{2}i \pi}=i$, and $e^{-\frac{1}{2}i \pi}=-i$. Therefore
$$\frac{e^{\frac{1}{2}i\pi}-e^{-\frac{1}{2}i\pi}}{2\pi i\frac{1}{2}}=\frac{2i}{2\pi i}=\frac{2}{\pi}$$
Carrying this correction through gives $1=1$ at the end.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4158071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Can there be a triangle ABC if $\frac{\cos A}{1}=\frac{\cos B}{2}=\frac{\cos C}{3}$? Can there be a triangle ABC if $$\frac{\cos A}{1}=\frac{\cos B}{2}=\frac{\cos C}{3}\;?$$ Equating the ratios to $k$ we get $\cos A=k$, $\cos B=2k$, $\cos C=3k$.
Then the identity $$\cos^2A+\cos^2B+\cos^2C+2\cos A \cos B \cos C=1 \impl... | \begin{align}
\frac{\cos A}{1}&
=\frac{\cos B}{2}=\frac{\cos C}{3}
=\frac{\cos A+\cos B+\cos C}{6}
\tag{1}\label{1}
.
\end{align}
\begin{align}
6\cos A&=\cos A+\cos B+\cos C
=\frac{r}{R}+1=
v+1
\tag{2}\label{2}
,
\end{align}
where $r$ and $R$ are inradius and circumradius
of the corresponding $\triangle ABC$,
$v=\tfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4158972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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About the inequality $x^{x^{x^{x^{x^x}}}} \ge \frac12 x^2 + \frac12$
Problem: Let $x > 0$. Prove that
$$x^{x^{x^{x^{x^x}}}} \ge \frac12 x^2 + \frac12.$$
Remark 1: The problem was posted on MSE (now closed).
Remark 2: I have a proof (see below). My proof is not nice.
For example, we need to prove that
$\frac{3x^2 - 3}... | Remarks: @Erik Satie considered $^6 x \ge \lim_{n\to \infty} {^n}x = -\frac{W(-\ln x)}{\ln x}$ for $(38/100, 1)$. I gave alternative
proof of $-\frac{W(-\ln x)}{\ln x} \ge \frac12 x^2 + \frac12$ for all $x$ in $(38/100, 1)$.
Case $x \in (38/100, 1)$:
According to Theorem in [1] (Page 240), we have
$\lim_{n\to \infty} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4165595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 2,
"answer_id": 1
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A six digit number is formed randomly using digits $\{1,2,3\}$ with repetitions. Choose the correct option(s):
A six digit number is formed randomly using digits $\{1,2,3\}$ with repetitions. Choose the correct option(s):
*
*A) Probability that all digits are used at least once is $\dfrac{20}{27}$
*B) Probability t... | +1 to your answer, for showing your work. I agree that your analysis for Part B is correct. My analysis is:
Part A
Use Inclusion-Exclusion.
$\displaystyle \frac{N\text{(umerator)}}{D\text{(enominator)}},~$ with $\displaystyle D = \frac{1}{3^6}.$
$\displaystyle N = 3^6 - \left[\binom{3}{1}2^6\right] + \left[\binom{3}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4166946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Basic algebraic/arithmetic manipulations I do not know the rules governing the transformation of each member of the following group of equations into the the next one:
$$ 5(1 + 2^{k -1} + 3^{k -1}) - 6(1 + 2^{k -2} + 3^{k -2}) + 2
\\= (5 -6 + 2) + (5 \times 2^{k - 1} - 6 \times 2^{k - 2}) + (5 \times 3^{k -1} - 6 \tim... | Here's an explanation of some of the lines:
*This line is obtained from line 1 by bringing the coefficients $5$, $6$ into their respective parentheses using the distributive law and then rearranging and regrouping terms using the commutative and associative laws. I assume there are no issues here, but if so, please a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4168488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Proving $\sum_{k=1}^{n}\cos\frac{2\pi k}{n}=0$ I want to prove that the below equation can be held.
$$\sum_{ k=1 }^{ n } \cos\left(\frac{ 2 \pi k }{ n } \right) =0, \qquad n>1 $$
Firstly I tried to check the equation with small values of $n$
$$ \text{As } n=2 $$
$$ \cos\left(\frac{ 2 \pi \cdot 1 }{ 2 } \ri... | The cosines are the $x$-coordinates of points on the unit circle equally spaced around the origin. Since the average $x$ value is zero, it follows that the sum of the $x$ values is also zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4170548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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If $3\sin x +5\cos x=5$, then prove that $5\sin x-3\cos x=3$
If $3\sin x +5\cos x=5$ then prove that $5\sin x-3\cos x=3$
What my teacher did in solution was as follows
$$3\sin x +5\cos x=5 \tag1$$
$$3\sin x =5(1-\cos x) \tag2$$
$$3=\frac{5(1-\cos x)}{\sin x} \tag3$$
$$3=\frac{5\sin x}{(1+\cos x)} \tag4$$
$$5\sin x-3\... | If $c=\cos x$ and $s=\sin x$, then you know that$$\left\{\begin{array}{l}3s+5c=5\\c^2+s^2=1.\end{array}\right.$$This system is easy to solve. One of the solutions is $(c,s)=(1,0)$ and the other one is $(c,s)=\left(\frac 8{17},\frac{15}{17}\right)$. In the first case (which is the case that you get when $x=0$), $5s-3c=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4171907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Show that $(a^2-b^2)(a^2-c^2)(b^2-c^2)$ is divisible by $12$
Let $a,b,c\in\Bbb N$ such that $a>b>c$. Then $K:=(a^2-b^2)(a^2-c^2)(b^2-c^2)$ is divisible by $12$.
My attempt : Since each $a,b,c$ are either even or odd, WLOG we may assume $a,b$ are both even or odd. For both cases, $a+b$ and $a-b$ are divisible by $2$ s... | To prove divisibility by $4$:
$(a^2-b^2)-(a^2-c^2)+(b^2-c^2)=0=\text{even}$
One of the addends must be even and this is possible with integers only if that is a multiple of $4$.
To prove divisibility by $3$:
Pigeonhole principle: at least two of $a^2,b^2,c^2$ must be multiples 9f $3$ or at least two must be one greater... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4172927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Showing$ \int_{0}^{\infty} \frac{1-3\cos 2x+2\cos 3x}{x^2} dx=0$ Showing $$\int_{0}^{\infty} \frac{1-3\cos 2x+2\cos 3x}{x^2} dx=0$$
We can show this by re-writing $I$ as
$$
\implies I=6\int_{0}^{\infty}\frac{\frac{1-\cos(2x)}{2x}-\frac{1-\cos(3x)}{3x}}{x}\,\mathrm dx,
$$
which is Frullani Integral.
$$J=\int_{0}^{\infty... | Alternatively, integrate by parts
\begin{align}
&\int_{0}^{\infty} \frac{1-3\cos 2x+2\cos 3x}{x^2} dx\\
= &\int_{0}^{\infty} \frac{6\sin 2x-6\sin 3x}{x} dx\\
=& \>6 \int_{0}^{\infty} \frac{\sin 2x}{2x} d(2x)
- 6 \int_{0}^{\infty} \frac{\sin 3x}{3x} d(3x) \\
=&\>0
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4179582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Let $α,β$ be the roots of $x^2-x+p=0$ and $γ,δ$ be the roots of $x^2-4x+q = 0$ where $p$ and $q$ are integers. If $α,β,γ,δ$ are in GP then $p + q$ is?
Let $α,β$ be the roots of $x^2-x+p=0$ and $γ,δ$ be the roots of
$x^2-4x+q = 0$ where $p$ and $q$ are integers. If $α,β,γ,δ$ are in GP
then $p + q$ is ?
My solution app... | The question can also be approached from the direction of working with the zeroes of the polynomials themselves. For $ \ x^2 - x + p \ = \ 0 \ $ and $ \ x^2 - 4x + q \ = \ 0 \ \ , $ we have
$$ \alpha \ , \ \beta \ \ = \ \ \frac{1 \pm \sqrt{1 - 4p}}{2} \ = \ \frac12 \ \pm \ \frac12 \sqrt{1-4p} \ = \ \frac12 \ \pm \ \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4180595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Solve : $(2x - 4y + 5) \frac{dy}{dx} + x - 2y + 3 = 0 , y(2) = 2.5$ I have to find the solution of following differential equation:
$$(2x - 4y + 5) \frac{dy}{dx} + x - 2y + 3 = 0 , y(2) = 2.5$$
It can be re-written as
$$(2(x - 2y) + 5) \frac{dy}{dx} + (x - 2y) + 3 = 0$$
let $u = (x-2y)$
so, on differentiating both si... | There's a mistake in following step and it is rectified in next step
$ \frac{1}{2}\cdot u - \frac{1}{8} \ln(8u + 22) = x + C$
$\implies \frac{1}{2}\cdot u - \frac{1}{8} \ln(|8u + 22|) = x + C$
$\implies \frac{1}{2}\cdot (x-2y) - \frac{1}{8} \ln(|8x-16y + 22|) = x + C$
When $x=2$ $y=2.5$, therefore
$\implies \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4180707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Given $f(r) = r\tan^{-1} \frac{a}{r+b}$ where $a>0$ and $b>0$, find an expression for $r$ in terms of $f(r)$. Is there an analytical solution to the following problem:
Given $f(r) = r\tan^{-1} \frac{a}{r+b}$ where $a>0$ and $b>0$ find an expression for $r$ in terms of $f(r)$.
To me, it seems like there is an analytical... | Around $r=0$, we can write the infinite sum for$$f=r \tan ^{-1}\left(\frac{a}{r+b}\right)$$ This gives
$$f=r \tan ^{-1}\left(\frac{a}{b}\right)+\sum_{n=1}^\infty \frac{(a-i b) (-b-i a)^{-n}+(a+i b) (-b+i a)^{-n}}{2 (n-1)}\, r^n$$ Truncate it to some order and, for more legibility, let $a=k b$
$$f=r \tan ^{-1}(k)-\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4181085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$ is divided by $(x^2 +1)$, then find the remainder
If the polynomial $x^{19}+x^{17}+x^{13}+x^{11}+x^{7}+x^{5}+x^{3}$
is divided by $(x^
2
+1)$, then the remainder is:
How Do I solve this question without the tedious long division?
Using remainder theorem , we can ta... | Following may help
$P(x)=x^{19} +x^{17} +x^{13} +x^{11} +x^{7} +x^{5} +x^{3}$
$P(x)=x^{17}(x^2+1)+x^{11}(x^2+1)+x^{5}(x^2+1)+x^3$
$P(x)=(x^2+1)(x^{17}+x^{11}+x^{5})+x^3$
That $x^3$ looks sad alone , Let's also include it
$P(x)=(x^2+1)(x^{17}+x^{11}+x^{5})+x^3+x-x$
$P(x)=(x^2+1)(x^{17}+x^{11}+x^{5}+x)-x$
$P(x)=(x^2+1)(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4182764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 6
} |
Mathematicals inequalities For $$x,y,z>0 $$
Prove that
$$(2xyz)^2 \ge (x^3+y^3+z^3+xyz)(x+y-z)(y+z-x)(z+x-y)$$
I have tried a famous inequalities: $$(x+y-z)(y+z-x)(z+x-y) \le xyz$$
So the problem is:
$$3xyz \ge x^3+y^3+z^3$$
But in fact, this isn’t true
Help me plz
| Suppose $z=\min\{x,y,z\}.$ We can write this inequality as
$$(x^3+y^3+z^3+xyz)\left [xyz-\prod (x+y-z) \right ] \geqslant xyz(x^3+y^3+z^3-3xyz),$$
But
$$xyz-\prod (x+y-z) = (x+y-z)(x-y)^2+z(x-z)(y-z),$$
$$x^3+y^3+z^3-3xyz = (x+y+z)[(x-y)^2+(x-z)(y-z)].$$
Thefore the inequality equivalent to
$$M(x-y)^2+N(x-z)(y-z) \geq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4186267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many points are common to the graphs of the two equations $(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$? How many points are common to the graphs of the two equations $(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$?
\begin{align*}
(x-y+2)(3x+y-4) &= 0\tag{1}\\
(x+y-2)(2x-5y+7) &= 0\tag{2}
\end{align*}
In equation $... | Compare equations indivually, and work case-by-case.
$$x-y+2=0\implies y=x+2, x+y-2=0\implies y=2-x$$
$$x+2=2-x\implies x=0\to y=2$$
So $(0,2)$ is a solution.
$$3x+y-4=0\implies y=4-3x,\\ x+2=4-3x\implies x=\frac12\to y=\frac52$$
So $(\frac12, \frac52)$ is a solution.
You can do the other two.
| {
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"url": "https://math.stackexchange.com/questions/4187238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Calculus Infinite Series Homework Problem. I have the problem
"Let
$f(x) = x + \frac{2}{3} x^3 + \frac{2 \cdot 4}{3 \cdot 5} x^5 + \dots + \frac{2 \cdot 4 \dotsm 2n}{3 \cdot 5 \dotsm (2n + 1)} x^{2n + 1} + \dotsb$
on the interval $(-1,1)$ of convergence of the defining series.
(a) Prove that $(1 - x^2) f'(x) = 1 + xf(x... | If you are given a power series $f(x) = \sum_{n = 0}^\infty a_n (x - x_0)^n$ with radius of convergence $R$, then $f$ is differentiable on $(x_0 - R, x_0 + R)$ and the derivative is $f'(x) = \sum_{n = 1}^\infty n a_n (x - x_0)^{n - 1}$.
In other words, you can differentiate term-by-term.
I shall leave it to you to veri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4187506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to solve $5^{x + 1} = 3^{x + 2}$
Solve $5^{x + 1} = 3^{x + 2}$.
I got this far, but I'm not sure how to continue:
\begin{align}
5^{x + 1} &= 3^{x + 2} \\
(5^x)(5^1) &= (3^x)(3^2) \\
5(5^x) &= 9(3^x)
\end{align}
Where do I go from here?
| Use natural logarithm:
$$5^{x+1}=3^{x+2}$$
$$\begin{align}&\implies (x+1)\ln 5=(x+2)\ln 3 \\
&\implies x\ln 5+\ln 5=x\ln 3+2\ln 3 \\
&\implies x(\ln 5-\ln 3)=2\ln 3-\ln 5 \\
&\implies x=\frac{2\ln 3-\ln 5}{\ln 5-\ln 3}. \end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4188828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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What is my mistake in finding this pythagorean triplet? Since Project Euler copyright license requires that you attribute the problem to them, I'd like to add that this is about question 9 there.
I am trying to solve this problem on only two brain cells and can't figure out what am I doing wrong. Here is the system for... | I think this comment by @MatthewLeingang explaining @lulu's comment answers the issue with my approach.
What lulu is saying by “not reversible” is that you have shown “If $a, b$, and $c$ are integers such that $a+b+c=1000$ and $a^2+b^2=c^2$, then $2c=1000− (ab/500)$.” That is not the same thing as “If $a$ and $b$ are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4191659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
We have $(x^2+y^2)^2-3(x^2+y^2)+1=0$. What is the value of $\frac{d^2y}{dx^2}$?
We have $(x^2+y^2)^2-3(x^2+y^2)+1=0$. What is the value of
$\frac{d^2y}{dx^2}$?
$1)-\frac{x^2+y^2}{y^2}\qquad\qquad2)-\frac{x^2+y^2}{y^3}\qquad\qquad3)\frac{x+y}{x^2+y^2}\qquad\qquad4)\frac{xy}{x^2+y^2}$
Here is my approach:
We have a qua... | Your approach is correct, you just made a small mistake at the last step: from $2+2y'^2+2yy''=0$ we find that
$$y''=-\frac{1+y'^2}{y}=-\frac{1+(-x/y)^2}{y}=-\frac{x^2+y^2}{y^3}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4194890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove $\sum_{n=2}^{\infty}\frac{1}{n^2+e}<\frac{1}{2}$ Of course, you can use the following formula
$$\sum_{n = 1}^\infty \frac{1}{n^2 + a^2} = \frac{\pi\coth(\pi a)}{2a} - \frac{1}{2a^2},$$
but which is too "advanced". We want to find a solution only depending on inequality estimation only.
Maybe, we can obtain
\begin... | This is a comment (I haven't enough reputation to comment).
Exact sum result:
$${{i\,\left(\psi_{0}(2-\sqrt{e}\,i)+\gamma\right)}\over{2\,\sqrt{e}
}}-{{i\,\left(\psi_{0}(\sqrt{e}\,i+2)+\gamma\right)}\over{2\,\sqrt{e
}}}$$
or
$$\frac{-3 e-1+(1+e) \pi \sqrt{e} \coth \left(\sqrt{e} \pi \right)}{2 e (1+e)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4195224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Computation of $\int_0^1 \frac{\arctan^2 x\ln x}{1+x}dx$ I'm searching for a "simple" proof of:
\begin{align}\int_0^1 \frac{\arctan^2 x\ln x}{1+x}dx=-\frac{233}{5760}\pi^4-\frac{5}{48}\pi^2\ln ^2 2+\text{Li}_4\left(\frac{1}{2}\right)+\frac{7}{16}\zeta(3)\ln 2+\frac{1}{24}\ln^4 2+\pi \Im\left(\text{Li}_3\left(1+i\right)... | Another framework proposed by Cornel (answer to the second integral, $\displaystyle \int_0^1\frac{\ln^2(1+x^2)\ln x}{1+x^2}\textrm{d}x$)
Observe that $$\int_0^1 \frac{1}{1+x^2}\log^3\left(\frac{2x}{1+x^2}\right)\textrm{d}x$$
$$=\log^3(2)\int_0^1\frac{1}{1+x^2}\textrm{d}x+3\log^2(2)\int_0^1\frac{\log(x)}{1+x^2}\textrm{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4196102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
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How many of the first $100$ terms are the same in the arithmetic sequences $2,9,16,\ldots$ and $5,11,17,\ldots$?
If $\{a_n\}$ is an arithmetic sequence with 100 terms where $a_1=2$ and $a_2=9$, and $\{b_n\}$ is an arithmetic sequence with 100 terms where $b_1=5$ and $b_2=11$, how many terms are the same in each sequen... | The general formulas for the terms are:
\begin{align*}
a_{n} = 2 + (n-1)7 = 7n - 5 \\
b_{n} = 5 + (n-1)6 = 6n - 1
\end{align*}
We are looking for two integers $x$ and $y$ such that:
\begin{align*}
a_{x} &= b_{y} \\
7x - 5 &= 6y - 1 \\
x &= \frac{6y + 4}{7} \\
\end{align*}
In other words we are looking a integer $y$ suc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4197691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Solve $\sqrt{\frac{(2 - x)(2 - y)}{(2 + x)(2 + y)} + \frac{(2 - y)(2 - z)}{(2 + y)(2 + z)} + \frac{(2 - z)(2 - x)}{(2 + z)(2 + x)}}$
Find the value of the expression $S$, knowing that $x^2 + y^2 + z^2 + xyz = 4$ and $x, y, z \in (0, \infty)$:
$$
S = \sqrt{\frac{(2 - x)(2 - y)}{(2 + x)(2 + y)} + \frac{(2 - y)(2 - z)}{... | One cannot find the value of $S$, only simplified expression.
Let's denote $x+y+z=a$, $xy+xz+yz=b$, $xyz=c$.
$x^2+y^2+z^2=a^2-2b \Rightarrow a^2-2b+c=4 \Rightarrow 2b=a^2+c-4$.
$S^2=A/B$, $A=(2-x)(2-y)(2+z)+(2-x)(2-z)(2+y)+(2-y)(2-z)(2+x)=$
$3xyz-2(xy+yz+xz)-4(x+y+z)+24=3c-2b-4a+24$,
$B=(2+x)(2+y)(2+z)=xyz+2(xy+yz+xz)+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4205951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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If $\phi(r) =1+\frac 12 +\frac 13… \frac 1r$ and $\sum_{r=1}^{n} (2r+1)\phi (r) =P(n)\phi(n+1)-Q(n)$. Find $P$ and $Q$. I tried making a double sum $$\sum_{r=1}^{n} \sum _{k=1}^r \frac{2r+1}{k}$$
But since the final limits aren’t same the changing of orders cannot be used. Can I get a hint on how to solve it?
| EDIT:
Using Python, I detected a small error in the final lines, which I have now fixed.
OP:
The sum
$$\sum_{r=1}^n\sum_{k=1}^r\frac{2r+1}{k}=3(1)+5\left(1+\frac{1}{2}\right)+7\left(1+\frac{1}{2}+\frac{1}{3}\right)+\dots+(2n+1)\left(1+\frac{1}{2}+\dots+\frac{2n+1}{n}\right)$$
Can be regrouped into sub-series as this:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4206092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $2^n + 5^n + 56$ is divisible by $9$, where $n$ is an odd integer
Prove that $9 \mid2^n + 5^n + 56$ where n is odd
I have proved this by using division into cases based on the value of $n\bmod3$ but it seems a little bit clumsy to me and I wonder if there are other ways to prove it, probably by using modul... | 2 times complete induction only $n\rightarrow n+1$
$2^n+5^n+56=9\cdot m\Rightarrow$
$2^{n+2}+5^{n+2}+56=4\cdot 2^n+25\cdot 5^n+56=3\cdot 2^n+24\cdot 5^n+2^n+5^n+56=3\cdot 2^n+24\cdot 5^n+9\cdot m$
$3\cdot 2^n+24\cdot 5^n+9\cdot m=3(2^n+8\cdot 5^n)+9\cdot m$
$2^n+8\cdot 5^n=3\cdot x$, because $2^{n+1}+8\cdot 5^{n+1}=2^n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4206716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 4
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Any trick for evaluating $\left(\frac{\sqrt{3}}{2}\cos(\theta) + \frac{i}{2}\sin(\theta)\right)^7$? Expressions of the form $(a\cos(\theta) + bi\sin(\theta))^n$ come up from time to time in applications of complex analysis, but to my knowledge the De Moivre's formula can only be applied with $a = b$. Is there some tric... | How about this:
$$ \frac{\sqrt{3}}{{2}} \cos \theta = \cos 30 \cos \theta= \frac{1}{2} \left[ \cos(30 - \theta)+\cos \left(30+ \theta \right) \right] =\frac12 (b+a)$$
$$ \frac12 \sin \theta= \sin(30) \sin \theta= \frac12 \left[\cos \left( 30 - \theta\right)- \cos (30+ \theta )\right]=\frac12 (b-a)$$
We have:
$$\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4206791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Finding $a$ such that $f(x)=a\,|x-b|$ is satisfied by $(2,1)$ and $(10,3)$
I was given the following equation: $$f(x)=a\,|x-b|$$ and the information that both $(2,1)$ and $(10,3)$ are solutions for the equation. The question asked to solve for $a$.
This is what I did: $$a=\frac{1}{2-b}\\[10pt]a=\frac{-1}{2-b}\\[10pt]... | I see two cases. $b\in(2,10)$ and $b\notin(2,10)$
$f(b) = 0$
In the case that $b$ is not in $(2,10)$ we construct the line through the two points. $b$ is the $x$-intercept of this line. $|a|$ is the slope.
$|a| = \frac {3-1}{10-2} = \frac 14$
Since this slope is postive, $a > 0, b < 2$
$0 = \frac 14 (x-2) + 1\\
0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4208529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Evaluate $\iiint_{V} (x^2+y^2+z^2)\,dx\,dy\,dz$ in the common part of $2az > x^2+y^2$ and $x^2+y^2+z^2 < 3a^2$ Evaluate $$\iiint_{V} x^2+y^2+z^2 \,dx\,dy\,dz$$
Where $V$ (the integration region) is the common part of the paraboloid $x^2 + y^2 \leq 2az$ and the sphere $x^2+y^2+z^2 \leq a^2$.
I first found the intercept ... | The lower bound of $z$ is correct. It should not be zero. However the order of $x$ and $y$ should be corrected - seems a typo.
But it is easier to do this in either cylindrical coordinates or spherical coordinates.
In cylindrical coordinates,
Sphere is $r^2+z^2 = 3 a^2$
Paraboloid is $2az = r^2$
At intersection, $2az +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4208625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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A JEE Exam problem on determinants and matrices I am first stating the question:
Let $A=\{a_{ij}\}$ be a $3\times 3$ matrix, where
$$a_{ij}=\begin{cases}
(-1)^{j-i}&\text{if $i<j$,}\\
2&\text{if $i=j$,}\\
(-1)^{i-j}&\text{if $i>j$,}
\end{cases}$$
then $\det(3\,\text{adj}(2A^{-1}))$ is equal to __________
I solved th... | No, your solution is not correct, but you are almost done.
Looking through the properties of adjugate matrix, we note that if $A$ is a $n\times n$ matrix then $\text{adj}(cA)=c^{n-1}\text{adj}(A)$ (not $c^n$ as you did) and $\text{adj}(A^{-1})=\det(A^{-1})A$. Therefore
$$\begin{align}\det(3\,\text{adj}(2A^{-1}))&=\det(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4213460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$a^a\cdot{b^b}\ge \bigl(\frac{a+b}{2}\bigl)^{a+b}\ge{a^b}\cdot{b^a}$ If $a$ and $b$ are positive rational numbers, prove that
$$a^a\cdot b^b\ge \left(\frac{a+b}{2}\right)^{a+b} \ge a^b \cdot{b^a}$$
My try:
consider $\frac{a}{b}$ and $\frac{b}{a}$ be two positive numbers with associated weights $b$ and $a$.
Then $\displ... | For the first inequality:
Let $f(x) = x\ln x$, then $f''(x) = 1/x > 0$ and hence convex. By the definition of a convex function, we have
$$f\left(\frac{a+b}{2}\right) \le \frac{f(a)+f(b)}{2}$$
which, in our case, becomes
$$\frac{a+b}{2} \ln \left(\frac{a+b}{2}\right) \le \frac{a\ln a + b\ln b}{2}$$
or equivalently
$$\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4214747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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The Diophantine equation $x^5-2y^2=1$ I'm trying to solve the Diophantine equation $x^5-2y^2=1$.
Here's my progress so far. We can write the Diophantine equation as
$$\frac{x-1}{2}\cdot(x^4+x^3+x^2+x+1)=y^2.$$
If $x\not\equiv1\pmod{5}$, then $\gcd(\frac{x-1}{2},x^4+x^3+x^2+x+1)=1$, so both $\frac{x-1}{2}$ and $x^4+x^3+... | Here is an "elementary" proof. The given diophantine equation $x^5 = 1+2y^2$ admits the obvious solution $x=1, y=0$. Exclude this trivial solution and consider $a=x^5$ as an integral parameter which one wants to represent as the value of the quadratic form $t^2+2y^2$, with unknown integers $(t,y)$. Geometrically, the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4218564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Can the formula for the cubes $a^3 + b^3 + c^3 - 3abc$ be generalized for powers other than 3? I recently learnt out about this formula:
$$a^3 + b^3 + c^3 - 3abc = (a + b + c) (a^2 + b^2 + c^2 - ab - ac - bc)$$
Is there a way of generalizing for powers other than $3$, i.e.
$$a^n + b^n + c^n + \mathop{???} = \mathop{???... | Too long for a comment:
One possible generalization is the circulant determinant, which tells that
$$ \det \begin{pmatrix}
a_1 & a_2 & a_3 & \cdots & a_n \\
a_n & a_1 & a_2 & \cdots & a_{n-1} \\
a_{n-1} & a_n & a_1 & \cdots & a_{n-2} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_2 & a_3 & a_4 & \cdots & a_1
\end{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4219021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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To prove: $\cot^{-1}7+\cot^{-1}8+\cot^{-1}18 = \cot^{-1}3$ To prove: $\cot^{-1}7+\cot^{-1}8+\cot^{-1}18 = \cot^{-1}3$
My Attempt:
First Method: we know that $\cot^{-1}x = \tan^{-1}\frac{1}{x}$ for $x>0$ and $\tan^{-1}x+\tan^{-1}y = \tan^{-1}\frac{x+y}{1-xy}, xy<1$
Now $\cot^{-1}7+\cot^{-1}8+\cot^{-1}18$ = $\tan^{-1}\fr... | ${tan^{-1}(1/7)+tan^{-1}(1/8)+tan^{-1}(1/18)}$
*
*${tan^{-1} \left[ \frac{ (1/7)+(1/8) } { \left( 1 \right) - (1/7)(1/8) } \right] + tan^{-1}(1/18) }$
*= ${tan^{-1} \frac {3}{11} + tan^{-1} \frac{1}{18}}$
*=${tan^{-1} \left[ \frac {(3/11)+(1/18)} {\left (1 \right) - (3/11)(1/18)} \right] }$
*= ${tan^{-1} \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4220297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Minimum value of algebraic expression This is a high school algebra problem,
If the minimum value of the expression $f(a,b) = \sqrt{a^2 + b^2 - 10a - 10b +50} +\sqrt{b^2 -4b +20} + \sqrt{a^2 - 14a +74}$ is $k$.
Which occurs at $a = \alpha$, $b = \beta$.
Find the value of $k + 4{\alpha} + 3\beta$
It can be seen that the... | Replace $a\to n+5$ and $b\to m+5$
Now the expression simplifies to $ \sqrt{n^2 + m^2} + \sqrt{ 4^2+(m+3)^2} + \sqrt{(n-2)^2 + 5^2}$
This is the sum of distance between the following pairs of points
*
*$(n,0)$ and $(2,5)$
*$(0,m)$ and $(n,0)$
*$(0,m)$ and $(-4,-3)$
If $m\ge0$,
we get $EA>EF$ and $FG+GD>AD$ (refer th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4229895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Proof by induction: induction hypothesis question In this question I found online:
*
*Show that $$ S(n):0^2 + 1^2 + 2^2 + · · · + n^2 =
\frac{n(n + 1)(2n + 1)}{6}$$
I don't understand why for S(k+1) they wrote:
$$S(k+1):1^2+2^2+3^2+⋯+k^2+(k+1)^2=\frac{(k+1)(k+2)(2(k+1)+1)}{6}$$
instead of:
$$S(k+1):1^2+2^2+3^2+⋯+(k+1... | Your mistake is in the last equation. It should read:
$$\frac{k(k+1)(2k+1)}{6} + (k+1)^2= \frac{(k+1)(k(2k+1)+6(k+1))}{6}= \frac{(k+1)(2k^2+7k+6)}{6} \\=\frac{(k+1)(k+2)(2k+3)}{6}=\frac{(k+1)((k+1)+1)(2(k+1)+1)}{6}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4231436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proof strategy for showing $a^2 + b^2 \neq c^2$ for $a,b$ odd, $c$ even. I am trying to understand a proof that $a^2 + b^2 = c^2$ cannot hold for $a,b$ odd and $c$ even.
The proof I am reading, and I believe I understand its steps, proceeds as follows. If $a,b$ are odd, then $a,b$ are congruent to either $1$ or $3$, mo... | Yes your reasoning is right.
For the first doubt note that for $a$ odd we have $a=4k+1$ or $a=4k+3$ and in any case since $a=2h+1$ we have $a^2=4h^2+4h+1\equiv 1 \pmod 4$.
For $c$ even we have that $c^2=4h^2\equiv 0 \pmod 4$.
Therefore we have
$$a^2+b^2=c^2 \implies a^2+b^2 \equiv c^2 \pmod 4$$
but as you noticed the l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4233068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Troubleshooting a trigonometry/geometry question - spot the (silly) mistake! I'm back again!
Again, another error - not sure if it's them or me this time...
Here is the question
In order to calculate the shaded region, I added some lines to the diagram. I realise that there are other ways to solve this (and indeed the... | The two answers are equivalent.
You have: $r^2[(\frac{9 \alpha}2)-9\sin(\frac{\alpha}2)+2 \theta]$
They have: $ [\frac 9 2(\alpha -\sin \alpha) + 2(\theta - \sin \theta)]r^2$
The triangle $ABD$ is isosceles, so $\frac \alpha 2 + 2 \frac \theta 2=\pi$
$\frac \alpha 2 = \pi-\theta \Rightarrow \sin \frac \alpha 2=\sin(\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4233346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How was the closed form of this alternating sum of squares calculated? I am reading through this answer at socratic.org.
The question is to find the closed form of the sum
$$1^{2}-2^{2}+3^{2}-4^{2}+5^{2}-6^{2}+\ldots.$$
I understand that, if the terms were added, the sum would be
$$
\sum_{n=1}^{N} n^{2}=1^{2}+2^{2}+\ld... | (This answer is an expansion of my comment. It shows an alternative method for computing the sum.)
If $n$ is even, then an easy way to compute $1^2-2^2+3^2-4^2+\dots+(n-1)^2+n^2$ is to consider the difference between each pair of terms:
\begin{align}
&1^2-2^2+3^2-4^2+\dots+(n-1)^2+n^2 \\[4pt]
=&(1-2)(1+2)+(3-4)(3+4)+\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4234536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to solve a limit that can be factored but doesn't help? I saw examples that can be factored, eliminating the part that causes the indetermination, none of this type. The other option is by rationalize but dont know how to apply it here.
$$\lim_{x \to 4} \frac{2x^2+7x+5}{x^2-16}$$
I tried by factoring, doesn't help
... | HINT
Your limit is in the form
$$\lim_{x \rightarrow 4} \frac{2x^2+7x+5}{(x-4)(x+4)} = \lim_{x \rightarrow 4} \frac{(2x+5)(x+1)}{(x+4)}\cdot \lim_{x \rightarrow 4} \frac{1}{x-4}$$
with
$$\lim_{x \rightarrow 4} \frac{(2x+5)(x+1)}{(x+4)} =\frac{65}{8}$$
then all boils down in that one
$$\lim_{x \rightarrow 4} \frac{1}{x-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4238283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determine the smallest $k$ such that $\log (1 +e^x) < k + x ?$ Determine the smallest $k$ such that $\log (1 +e^x) < k + x $ where $k$ is constant and $x \in (0,\infty)$
My attempt :$$\log(1+e^x)= e^x -\frac{e^{x}}{2} +\frac{e^{2x}}{3}+.....$$
$$e^x = 1 +x+x^2/2^2+...$$
$$\log(1+e^x)= (1+x+x^2/2^2+....)-\frac... | $f(x)=e^x (e^k -1)>1$ is strictly increasing.
$\\ $ $f(x\to +0)>1 \implies k>\ln 2 $ or $\inf k =\ln 2 $.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4239898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Finding the inverse of a block $2\times2$ square matrix $ \begin{bmatrix} I & A \\ A^T & 0 \end{bmatrix} $ where $A$ is a square invertible matrix. The hint I got was to rewrite the original matrix as a product of 3 matrices and use the property for inverse of product of matrices $(XYZ)^{-1} = Z^{-1} Y^{-1} X^{-1}$
$\b... | Ben Grossmann already pointed out that you have a flaw in the product. You can't factor invertible matrix into a product of matrices some of which are not invertible. This is an easy consequence of Binet-Cauchy formula: $\det(AB) = \det A\cdot\det B$.
There is an approach that doesn't require anything smart to find the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4240327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Find the limit $\lim_{x \to 0}{\left(\frac{\tan x}{x}\right)}^{1/{x^2}}$ using l'Hôpital's Rule. Here is the limit I'm trying to find.
$$\lim_{x \to 0}{\left(\frac{\tan x}{x}\right)}^{1/{x^2}}$$
Now, since it takes an indeterminate form at $x=0$, I'm taking natural logarithm and trying to find the following limit.
$$
\... | For this type of problem, it is simpler to get the limit using the Maclaurin series of the tangent function and the logarithm function. Letting $z = 1/x^2$ we can write:
$$\begin{align}
\frac{1}{x^2} \log \bigg( \frac{\tan x}{x} \bigg)
&= \frac{1}{x^2} \Bigg[ \log \tan (x) - \log(x) \Bigg] \\[6pt]
&= \frac{1}{x^2} \Bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4242272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Intuition behind getting two straight lines as result Question:
Find the equation of the straight line that passes through $(6,7)$ and makes an angle $45^{\circ}$ with the straight line $3x+4y=11$.
My solution (if you want, you can skip to the bottom):
Manipulating the given equation to get it to the slope-intercept fo... | You have a right triangle with the right angle at (6, 7) and the line 3x+ 4y= 11 as hypotenuse. The "two lines" are the two legs of the triangle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4244257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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A square root of −4 modulo the ideal generated by $(x^2 + 1)^2$ Let $S= \mathbb R[x]$ denote the polynomial ring in one variable over the field $\mathbb R$ of real numbers. Find a
monic polynomial of least degree in $S$ that is a square root of $−4$ modulo the ideal $I$ generated by
$(x^2 + 1)^2$.
Let $p(x)$ be a monic... | The basis $1,x,x^2,x^3$ has no relation to $(x^2+1)^2$, which is the polynomial which you are trying to go modulo. Instead, what is usually done is traditionally called "lifting" as suggested in the comments by Jyrki. (So it wasn't misread, it was a hint).
The idea of lifting is simple : if an equation is true modulo $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4244907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Prove that $\frac{1}{2(n+2)}<\int_0^1\frac{x^{n+1}}{x+1}dx$ $\displaystyle\int\limits_0^1\frac{x^{n+1}}{x+1}dx=\left[\frac{x^{n+2}}{(n+2)(x+1)}\right]_0^1+\int\limits_0^1\frac{x^{n+2}}{(x+1)^2(n+2)}dx$
$\displaystyle\int\limits_0^1\frac{x^{n+1}}{x+1}dx=\frac{1}{2(n+2)}+\int\limits_0^1\frac{x^{n+2}}{(x+1)^2(n+2)}dx$
If ... | If the function you are integrating is always positive, then the integral must be positive as well. Take a look at the function being integrated. We have
$$f(x) = \frac{x^{n+2}}{(x+1)^2(n+2)}$$ with $0\leq x \leq 1$. Given that $x\in [0,1]$, is it true that $f(x)$ is always positive?
I assume also that you mean to find... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4245166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Independent chances of $3$ events ${a\over{a+x}}$, ${b\over{b+x}}$, ${c\over{c+x}}$ Here's a problem from my probability textbook:
Of three independent events the chance that the first only should happen is a; the chance of the second only is $b$; the chance of the third only is $c$. Show that the independent chances ... | To get corner cases out of the way, if $p_1=1$ then $b=c=0$, and $x=1-a$ will satisfy all the wanted properties. If $p_1=0$, then $a=0$ and the problem reduces to a similar one with only two events. The same goes for $p_2$ and $p_3$, so from here on assume $0<p_1<1$, $0<p_2<1$, and $0<p_3<1$.
From your observation
$$a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4245883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
What is the value of the $\measuredangle IEP$ in the figure below? In a right triangle ABC straight at B, $\measuredangle C = 37^\circ. $If $E$ is an excenter in relation to BC, I is an incenter and $P$ is the point of tangency of the circle inscribed with AC.
calculate $\measuredangle IEP$
My progres::
I made the draw... | HINT:
$\small{\triangle ABC}$ is a $\small{3,4,5}$ triangle and $\small{\triangle AJE}$ is a right triangle with perpendicular sides in the ratio $\small{1:2}$. Therefore say $\small{AB=3, BC=4, AC=5}$ and $\small{BJ=JE=3}$. This leads $\small{AE=3\sqrt 5}$.
Let $\small{AB}$ touches incircle at $\small Q$, then define ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4251567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.