Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Generating function - technical issue. Which sequence is generated by $\frac{{5x - 3{x^2}}}{{{{(1 - x)}^3}}}$?
We know that:
$$\frac{1}{{{{(1 - x)}^3}}} = \sum\limits_{j = 0}^\infty {\left( {\begin{array}{*{20}{c}}
{j + 2} \\
2 \\
\end{array}} \right){x^j}} $$
So we have:
$$(5x - 3{x^2}) \cdot \sum\limits_... | I think you're overthinking things. First of all,
\begin{align*}
5 {k + 1 \choose 2} - 3{k \choose 2}
&= 5\frac{(k+1)k}{2} - 3\frac{k(k-1)}{2} \\
&= \frac{5}{2} k^2 + \frac52 k - \frac32k^2 + \frac32 k \\
&= 4k + k^2 \\
&= k(k + 4) \\
\end{align*}
So your sequence is $a_k = k(k+4)$. Second of all, they probably want ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/703482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
The normal line intersects a curve at two points. What is the other point? The line that is normal to the curve $\displaystyle x^2 + xy - 2y^2 = 0 $ at $\displaystyle (4,4)$ intersects the curve at what other point?
I can not find an example of how to do this equation. Can someone help me out?
| Given the curve $\mathcal{C}$ as
$$
x^2 + x y - 2 y^2 = 0. \tag 1
$$
Solve the curve $\mathcal{C}$
$$
x^2 + x y - 2 y^2 = 0.\\
\Downarrow\\
4 x^2 + 4 x y - 8 y^2 = 0.\\
\Downarrow\\
\Big( 2 x + y \Big)^2 - 9 y^2 = 0.\\
\Downarrow\\
\Big( 2 x + y \Big)^2 = \Big( 9 y \Big)^2.\\
\Downarrow\\
2 x + y = \pm 3 y.\\
\Downar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/707716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Definite integration with natural logarithm $$\int_0^a \ln(x)\ln(a-x)\,dx$$
How to do this? I couldn't proceed at all.
($\ln$ is natural logarithm.)
| $a>0$
\begin{align}J&=\int_0^a \ln(x)\ln(a-x)\,dx\\
&=\frac{1}{2}\int_0^a \ln^2(x)dx+\frac{1}{2}\int_0^a \ln^2(a-t)dt-\frac{1}{2}\int_0^a \ln^2\left(\frac{u}{a-u}\right)du\\
&\overset{x=a-t,x=\frac{u}{2-u}}=\int_0^a \ln^2(x)dx-\frac{a}{2}\int_0^\infty \frac{\ln^2 x}{(1+x)^2}dx\\
&=\int_0^a \ln^2(x)dx-\frac{a}{2}\int_0^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/708072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Inverse trigonometric Problem For any $x \in [-1,0) \cup (0,1]$, how can I prove that:
$$\sin^{-1}(2x\sqrt{1-x^2})=2\cos^{-1}x$$
Also, can someone explain to me how to understand the graphs of $sin$ and $cos$ functions?
| Sufficient care has to be taken while dealing with Inverse trigonometric functions
as the principal values of $\cos^{-1}x$ lies in $\left[0,\pi\right]$ whereas it lies in $\left[-\frac\pi2,\frac\pi2\right]$ for $\sin^{-1}y$
$\displaystyle\iff0\le2\cos^{-1}x\le2\pi$
Case $\#1:\displaystyle2\cos^{-1}x$ will be $\display... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/709863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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What is wrong with this method for a rotated and shifted parabola? $(x+2y)^2=4(x-y)$
Disecting the above parabola is the question. (vertex, axis,tangent at vertex,etc).
So at first what I thought of was making its equations at LHS and RHS perpendicular. I thought since it might be tedious (which wasnt at all and got me... | This is actually the
duplicate of an answer
to the following question:
Proving that for each two parabolas, there exists a transformation taking one to the other .
According to the theory presented there, it is advantageous to write the OP's equation as follows:
$$
2(x-y) = \frac{1}{2} (x+2y)^2 \qquad \mbox{or} \qquad ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/711004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove by induction that $\sum_{i=0}^n \left(\frac 3 2 \right)^i = 2\left(\frac 3 2 \right)^{n+1} -2$
Prove, disprove, or give a counterexample:
$$\sum_{i=0}^n \left(\frac 3 2 \right)^i = 2\left(\frac 3 2 \right)^{n+1} -2.$$
I went about this as a proof by induction. I did the base case and got the LHS = RHS. When I w... | The inductive step:
$$\begin{align}\sum_{i=0}^{k+1}\left(\frac{3}{2}\right)^{i} &= \left(\frac{3}{2}\right)^{k+1} + \sum_{i=0}^{k}\left(\frac{3}{2}\right)^{i}\\
&=\left(\frac{3}{2}\right)^{k+1} + 2\cdot\left(\frac{3}{2}\right)^{k+1} - 2\\
&= 3\cdot\left(\frac{3}{2}\right)^{k+1} - 2\\
&= 3\cdot\frac{2}{3}\left(\frac{3}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/712707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Length of median extended to the circumcircle A triangle has side length $13,14,15$, and its circumcircle is constructed. The median is then drawn with its base having a length of $14$, and is extended to the circle. Find its length.
|
Here, $AB=13, AC=15, BC=14$. $AD$ is the median on $BC$ and we have to find the length of $AE$. So, we first take out the length of $m=AD$. Noting by law of cosines that:
$$m^2=c^2-(\frac{a}2)^2+am\cos \alpha$$
$$m^2=b^2-(\frac{a}2)^2+am\cos \beta=b^2-(\frac{a}2)^2-am\cos \alpha$$
And adding them up, to get the Apollo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Finding the common ratio from adjacent elements in a geometric sum Is there a general solution for this problem?
Given $S_N$ and $S_{N+1}$ are the sums of geometric series, can we find the common ratio(s). (Assuming $a_0 = 1$).
To be more explicit, given
$$S_N = 1 + r + \ldots + r^{N-1} = \frac{1-r^N}{1-r}$$
and
$$S_{N... | We have
$$S_{N+1}-1=r+r^2+\cdots+r^N=rS_N\ .$$
So if $S_N$ and $S_{N+1}$ are given you can easily find
$$r=\frac{S_{N+1}-1}{S_N}\ .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/716319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Trigonometric limit $\lim_{x\to0} \frac{\tan^2{(3x)}+\sin{(11x^2)}}{x\sin{(5x)}}$ How to solve this limit:
$$\lim_{x \rightarrow 0}\frac{\tan^2{(3x)}+\sin{(11x^2)}}{x\sin{(5x)}}$$
| write
$$\frac{\tan^2(3x)+\sin(11x^2)}{x\sin 5x}=\frac{3}{1}\cdot\frac{3}{1} \cdot \frac{1}{5}\cdot \frac{\tan 3x}{3x}\cdot \frac{\tan 3x}{3x}\cdot \frac{5x}{\sin 5x}+ \frac{\sin (11x^2)}{11x^2}\cdot \frac{5x}{\sin 5x}\cdot \frac{1}{5}\cdot \frac{11}{1}$$
Making the passage to the limit when $x\rightarrow 0$ each fracti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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What is the smallest positive integer n such that $2^5 · 3 · 5^2 · 7^3 ·$ $n$ is a perfect square Is there some kind of rule that I need to solve this? Can someone give me some clue how to solve this? Thanks
My teacher gave the solution as $42$.
Can someone explain why?
| I will use the rule that:
$$a^2b^2=(ab)^2$$
We want all exponents to be even. Why? Let's use an example.
(Assume all variables are integers, and are not perfect squares themselves).
$a^2b^4$ is a perfect square because it equals $(ab^2)^2$. But $a^2b^3$ is not a perfect square because there is no way to convert it into... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/721845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Evaluate limit $\lim_{n \to \infty } {1 \over n^{k + 1}}\left( {k! + {(k + 1)! \over 1!} + \cdots + {(k + n)! \over n!}} \right),k \in \mathbb{N}$ Evaluate the limit:
$$\lim_{n \to \infty } {1 \over n^{k + 1}}\left( {k! + {(k + 1)! \over 1!} + \cdots + {(k + n)! \over n!}} \right),k \in \mathbb{N}$$
It looks like a cla... | Hint: It can be shown that
$$\sum_{m = 0}^n \frac{(k + m)!}{m!} = \frac{(n + 1)(k + n + 1)!}{(k + 1)(n + 1)!}.$$
You may find this formula useful.
If you then divide by $n^{k + 1}$ and take the limit as $n \to \infty$, you should get
$$\frac{1}{1 + k}.$$
Edit: Using Stirling's asymptotic formula $N! \sim N^N e^{-N}\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/722621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Compute the square root of a complex number This is a follow up to a previous question. I solved the equation $z^4 - 6z^2 + 25 = 0$ and I found four answer to be $z = \pm\sqrt{3 \pm 4i}$.
However someone in the comment said that the answer is going to be $2+i$, $2-i$, $-2+i$, $-2-i$. I cannot understand how we can fin... | Suppose $(x+iy)^2 = a+ib$ with $a,b$ real and you want to find real values for $x$ and $y$.
Then $(x^2-y^2) + i(2xy) = a+ib$. Since $a,b,x,y$ are real, this is equivalent to $x^2-y^2 = a$ and $2xy = b$
$(x^2+y^2)^2 = (x^2-y^2)^2 + 4x^2y^2 = a^2+b^2$, so $x^2+y^2 = \sqrt {a^2+b^2}$. You always pick the positive root bec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/722852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Specify finite group satisfying two conditions Let $G = \left\{ 0, 1, 2, 3, 4, 5, 6, 7\right\}$ be a group with the operation $\circ$, which satisfies the following conditions:
\begin{align}
a \circ b \leq a + b \quad & \forall a,b \in G & (a) \\
a \circ a = 0 \quad & \forall a \in G & (b)
\end{align}
I have to create ... | There is a shorter way to solve the problem, although I wouldn't call it elegant.
But before that, the more important thing: there is also a very short way to describe how this operation $\circ$ that you found operates. To calculate $a \circ b$, you just write $a$ and $b$ in binary, then apply the bitwise XOR, and then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/725128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Question Regarding Telescopic Sums How do I find a formula for the partial sum $s_n$ for the series $$\sum_{k=1}^{\infty}(\frac{1}{k}-\frac{1}{k+2})$$
| For example: $$\left(\dfrac{1}{1} - \dfrac{1}{3}\right) + \left(\dfrac{1}{2} - \dfrac{1}{4}\right) + \left(\dfrac{1}{3} - \dfrac{1}{5}\right) =
\dfrac{1}{1} + \dfrac{1}{2} - \dfrac{1}{4} - \dfrac{1}{5}$$
Try a few more: do you see the pattern?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/726254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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How to prove for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$? I'm new to induction so please bear with me. How can I prove using induction that, for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$?
I think $9$ can be an example since the sum of t... | Let $P(n)$ be the statement that the sum of the first $n$ odd positive integers is $n^2$. $1=1^2$, so we have $P(1)$.
Suppose we have $P(k)$ for some positive integer $k$. Then the sum of the first $k$ odd positive integers is $k^2$. Now the $(k+1)$th odd positive integer is $2k+1$, so the sum of the first $k+1$ odd po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/727774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 11,
"answer_id": 6
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One Diophantine equation I wonder now that the following Diophantine equation:
$2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$
have only this formula describing his decision?
$a=-(k^2+2(p+s)k+p^2+ps+s^2)$
$b=2k^2+4(p+s)k+3p^2+3ps+2s^2$
$c=3k^2+4(p+s)k+2p^2+ps+2s^2$
$d=2k^2+4(p+s)k+2p^2+3ps+3s^2$
$k,p,s$ - what some integers.
By your ... | The above equation which is mentioned below has another parametrisation,
$2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$
$a=(5k+2)^2$
$b=(3k-4)^2$
$c=(2k+6)^2$
$d=4(19k^2+10k+28)$
For $k=0$ we get $(a,b,c,d)=(1,4,9,28)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/728994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Analyzing a fourth degree polynomial Let $a,b$ and $c$ be real numbers. Then prove that the fourth degree polynomial in $x$
$acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac$ has either 4 real roots or 4 complex roots.
I have never solved a fourth degree polynomial and don't know the conditions for it to have real/complex ... | Notice that the given expression,
$$acx^4+b(a+c)x^3+(a^2+b^2+c^2)x^2+b(a+c)x+ac = 0$$
can be factorized into (see below for derivation):
$$(ax^2 + bx + c)(cx^2 + bx + a) = 0$$
The discriminant for each is the same, $b^2 - 4ac$.
If this common discriminant is zero or more, then the roots for both $ax^2 + bx + c = 0$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Integral $I=\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx$ Hi I am stuck on showing that
$$
\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx=\pi G-\frac{3\zeta(3)}{8}
$$
where G is the Catalan constant and $\zeta(3)$ is the Riemann zeta function. Explictly they are given by
$$
G=\beta(2)=\sum_{n=0}^\infty \frac{(-1... | The infinite sum in Chen Wang's answer, that is, $ \displaystyle \sum_{n=1}^{\infty} \frac{H_{4n}}{n^{2}}$, can be evaluated using contour integration by considering the function $$f(z) = \frac{\pi \cot(\pi z) [\gamma + \psi(-4z)]}{z^{2}}, $$
where $\psi(z)$ is the digamma function and $\gamma$ is the Euler-Mascheroni ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/730732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 1
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How do I integrate $\frac{1}{x^6+1}$ My technique so far was substitution with the intent of getting to a sum of three fractions with squares in their denominators.
$t = x^2 \\
\frac{1}{x^6 + 1} = \frac{1}{t^3+1} = \frac{1}{(t+1)(t^2-t+1)}$
Then I try to reduce this fraction into a sum of two fractions
$\frac{A}{t+1} +... | $$\begin{align}x^6+1=(x^2)^3+1^3&=(x^2+1)(x^4-x^2+1)=(x^2+1)((x^2+1)^2-3x^2)\\&=(x^2+1)(x^2-\sqrt3\,x+1)(x^2+\sqrt3\,x+1)\end{align}$$
Now you can use partial fractions:
$$\frac1{(x^2+1)(x^2-\sqrt3\,x+1)(x^2+\sqrt3\,x+1)}=\frac{Ax+B}{x^2+1}+\frac{Cx+D}{x^2-\sqrt3\,+1}+\frac{Ex+F}{x^2+\sqrt3\,+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/734399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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How to integrate $\int_{1}^{3} {\frac{x^2 - 1}{x^4 + 1}}\, dx$ $$\int_{1}^{3} {\frac{x^2 - 1}{x^4 + 1}}\, dx$$
Well, I can simplify the numerator:
$$\int_{1}^{3} {\frac{(x - 1)(x+1)}{x^4 + 1}}\, dx$$
But I have no idea how to simplify the denumenator:
$$\ x^4+1=? \ $$
How to solve this integral?
| $$
\begin{align}
\hspace{-1cm}\int_1^3\frac{x^2-1}{x^4+1}\mathrm{d}x
&=\int_1^3\frac{x^2-1}{(x^2+1)^2-2x^2}\mathrm{d}x\tag{1}\\
&=\int_1^3\left[\frac{-1/2+x/\sqrt2}{x^2-\sqrt2x+1}
+\frac{-1/2-x/\sqrt2}{x^2+\sqrt2x+1}\right]\mathrm{d}x\tag{2}\\
&={\Large\int}_1^3\left[\frac{\frac1{\sqrt2}\left(x-\frac1{\sqrt2}\right)}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/736897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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What converges this series? What exactly converges the series?
$\sum _{k=3}^{\infty \:}\frac{2}{k^2+2k}$
I tried taking out the constant
$=2\sum _{k=3}^{\infty \:}\frac{1}{k^2}$
then $p=2,\:\quad \:p>1\quad \Rightarrow \sum _{k=3}^{\infty \:}\frac{1}{k^2}$
but I really don't know what i'm doing
| Use partial fractions: Suppose
$$f(n)=\sum\limits_{k=3}^n\dfrac{2}{k^2+2k}=\sum\limits_{k=3}^n(\dfrac{1}{k}-\dfrac{1}{k+2})$$
$$=(\frac{1}{3}-\frac{1}{5})+(\frac{1}{4}-\frac{1}{6})+(\frac{1}{5}-\frac{1}{7})+\dots+(\frac{1}{n-2}-\frac{1}{n})+(\frac{1}{n-1}-\frac{1}{n+1})+(\frac{1}{n}-\frac{1}{n+2})$$
All the terms cance... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/737816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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How to prove $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$ Question:
Let $a,b,c>0$ are give numbers and $x>0$, such that
$$
\sqrt{\dfrac{a+b+c}{x}}=\sqrt{\dfrac{b+c+x}{a}}+\sqrt{\dfrac{c+a+x}{b}}+\sqrt{\dfrac{a+b+x}{c}}
$$
show that
$$
\dfrac{1}{x}=\dfrac{1}{a}+\d... | You can prove this equation using geometry approach. Note that this equation is exactly similar with Descartes 4 Circles Theorem. Descartes' theorem says: If four circles are tangent to each other at six distinct points and the circles have curvatures $k_i$ (for $i = 1,\cdots, 4$), then $k_i$ satisfies the following re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
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What is the value of $\int x^x~dx$? I am struggling with this puzzle.
Question 1. Is it possible to determine the value of the indefinite integral $\int x^x~dx$ explicitly?
By "explicit" I mean without power series.
Question 2. What are known theorems in the following form?
"$\int x^x~dx$ is not expressible by an expre... | $\int x^x~dx=\int e^{x\ln x}~dx=\int\sum\limits_{n=0}^\infty\dfrac{x^n(\ln x)^n}{n!}dx$
For $\int x^n(\ln x)^n~dx$ , where $n$ is any non-negative integers,
$\int x^n(\ln x)^n~dx$
$=\int(\ln x)^n~d\left(\dfrac{x^{n+1}}{n+1}\right)$
$=\dfrac{x^{n+1}(\ln x)^n}{n+1}-\int\dfrac{x^{n+1}}{n+1}d((\ln x)^n)$
$=\dfrac{x^{n+1}(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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Integrals Clarification Needed
Missed a couple days of class, can't seem to figure out how to get the numbers marked with the blue arrows. Other than that I understand the concept.
| Alright, I've done this problem on ms paint. This is how the answer is $-136$
We have the definite integral of
$\int^4_2 (10+4x-3x^3)\,dx$
Before we can evaluate the definite integral, we need to take the antiderivative of $10+4x-3x^3$. To take the antiderivative we need to add one to the exponent and divide by the ne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/742807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is the derivative of circle area the circumference? Why is the derivative of the volume of a sphere the surface area? And why is the derivative of the area of a circle the circumference? Too much of a coincidence, there has to be a reason!
Also, why is the sum of the cubes the square of the sum of the integers?
| I'll answer the last question first because that's the easiest one in my opinion.
Prove that
$$1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
We will prove this using induction, by first confirming the identity when $n = 1$ and then proving the inductive step.
Base Case: $n = 1$
$$1^3 = \left(\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/744600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Let $u_{n+3} = u_n + 2u_{n+1}$ . Show that $p$ divides $u_p$ for all $p$ prime number.
Let $(u_n)$ a sequence such that $u_0 = 3$, $u_1 = 0$, $u_2 = 4$ and $u_{n+3} = u_n + 2u_{n+1}$
Show that $p$ divides $u_p$ for all $p$ prime number.
I'm really stuck on this exercise,
Does anyone can give me a good HINT to start... | The characteristic equation for the recurrence relations $u_{n+3} - 2u_{n+1} - u_n = 0$ is
given by
$$\lambda^3 - 2\lambda - 1
= (\lambda-1)\left(\lambda-\frac{1+\sqrt{5}}{2}\right)\left(\lambda-\frac{1-\sqrt{5}}{2}\right)$$
Since the roots are all simple, the general solution for $u_n$ has the form
$$u_n = \alpha (-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/746646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Probability of eight dice showing sum of 9, 10 or 11 Suppose we roll eight fair dice. What is the probability that:
1) The sum of the faces is $9$
2) The sum of the faces is $10$
3) The sum of the faces is $11$
I'm thinking that we start with $8$ dice, each showing $1$. Then we think of the problem as assigning one $1$... | In all possible orders:
1) $21111111$. Probability: $\frac{8!}{1!7!}\times6^{-8}$
2) $31111111$ or $22111111$. Probability: $\left(\frac{8!}{1!7!}+\frac{8!}{2!6!}\right)\times6^{-8}$
3) $41111111$or $32111111$ or $22211111$. Probability: $\left(\frac{8!}{1!7!}+\frac{8!}{1!1!6!}+\frac{8!}{3!5!}\right)\times6^{-8}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/746970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$
Prove that positive $x,y,z$ satisfy $$\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$$
Actually, this is a part of my solution to another problem, which is:
If $a,b,c$ are sides of a triangle, p... | \begin{align}
& 2(\sqrt{x+y}+\sqrt{x+z}+\sqrt{y+z}) \geq 2(\sqrt{2x}+\sqrt{2y}+\sqrt{2z}) \\
& \Leftrightarrow (2\sqrt{x+y}-\sqrt{2x}-\sqrt{2y})+(2\sqrt{x+z}-\sqrt{2x}-\sqrt{2z})+(2\sqrt{y+z}-\sqrt{2y}-\sqrt{2z}) \geq 0
\end{align}
Note that
\begin{align}
& (2\sqrt{x+y}-\sqrt{2x}-\sqrt{2y}) \geq 0 \\
& \Leftrightarro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/748866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
The general term formula $a_{n+1}=\frac{1+a_n^2}2$ Let $\ a_1=\dfrac 12\ $ and $\ a_{n+1}=\dfrac{a_n^2+1}2$,
*
*Could we find the general term formula $a_n$?
*If the answer to the question $1$ is "NO", for $\left|a_n-\dfrac{n-1}{n+1}\right|$ and $\left|a_n-\dfrac{n}{n+1}\right|$, which one is small in comparison w... | (1) As it was noted in the comments above, the sequence $(a_n/2)_{n\geq1}$ is defined by the iteration of the function $f(x)=x^2+c$ with $c=1/4$ and consequently it does not have a closed form.
(2) By a simple induction we see that $\frac{1}{2}\leq a_n<a_{n+1}<1$ for each $n$, This allows us to conclude that $(a_n)$ co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/749813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to simplify this trignometric expression: $4( 3 \sin \theta)( 3 \cos \theta)$? I was given a circle with a radius of $3$ and in it was a rectangle and an angle $\theta$ extending from the $x$ axis to up with coordinates of $(3 \cos \theta, 3 \sin \theta)$ and the question asks me to show that the area of the triang... | $\text{Area}= \frac 1 2 3\sin\theta 3\cos\theta$
Use $\sin\theta \cos\theta = \frac 1 2\sin 2\theta$
$=\frac 9 4 \sin 2\theta$
Multiply by $8$ to get $18\sin 2\theta$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/752029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $a+b$ is a multiple of $24$? Let $x$ be an integer one less than a multiple of $24$. Prove that if $a$ and $b$ are positive integers such that $ab=x$, then $a+b$ is a multiple of $24$.
| So $ab \equiv -1 \pmod {24} \implies \gcd(a,24) = \gcd(b,24) = 1$. Then let $ab = 24k-1$ so $b = \frac{24k-1}a$ and $a+b = \frac{24k-1+a^2}a$. Since $\gcd(a,24) = 1$, it follows that $a^2 \equiv 1 \pmod {24}$ and we are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/754336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
How do I solve $(x^3-4x^2+5x-6)/(x^2-x-6)=4$ algebraically? How do I solve $\frac {(x^3-4x^2+5x-6)}{(x^2-x-6)=4}$ algebraically?
I tried:
$4(x^2-x-6)=x^3-4x^2+5x-6$
$4x^2-4x-24=x^3-4x^2+5x-6$
$x^3-8x^2+9x+18=0$
I don't know how to solve this algebraically.
| $$x^3-4x^2+5x-6 = (x-3)(x^2-x+2)$$
$$x^2-x-6 = (x-3)(x+2)$$
so the equation becomes: $$\frac{x^2-x+2}{x+2} =4 $$
$$x^2-x+2=4x+8$$
$$x^2-5x-6=0$$
$$(x-6)(x+1)=0$$
$x=6$ or $-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/755084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Cayley-Hamilton Block Matrix In Q4 d) I understand how to get the expression for A (using Q3), but I don't understand how you can then say that C is of the form described.
| Well, because $C$ is a block diagonal matrix, we have
$$ C^² + C + I_4 = \begin{pmatrix}
A^² & 0 \\
0 & A^² \\
\end{pmatrix} + \begin{pmatrix}
A & 0 \\
0 & A \\
\end{pmatrix} + \begin{pmatrix}
I_2 & 0 \\
0 & I_2 \\
\end{pmatrix} $$
which si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/755158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Estimation of integral Suppose the function $f(x)$ has a Taylor series expansion. Then
$$\int_a^bf(x)dx=\int_a^b(f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2+\cdots)dx=\\ \frac{f(a)}{1!}(b-a)+\frac{f'(a)}{2!}(b-a)^2+\frac{f''(a)}{3!}(b-a)^3+\cdots$$
and
$$\int_a^bf(x)dx=\int_a^b(f(b)+f'(b)(x-b)+\frac{1}{2}f''(b)(x-b)^2+\cd... | Remarkably, they appear to both be correct. I have checked the following integrals using both formulas: $$\int_0^{\pi/2}\sin x\,\mathrm{d}x=1,\;\;\int_0^{\pi/2}\cosh x\,\mathrm{d}x\approx 2.301,\;\;\int_a^{b}x^2\,\mathrm{d}x=\frac{b^3-a^3}{3},\;\;\int_a^{b}x^3\,\mathrm{d}x=\frac{b^4-a^4}{4}$$
See for example
Here and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/755565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
If $d=\gcd(a+b,a^2+b^2)$, with $\gcd(a,b)=1$, then $d=1$ or $2$ Suppose $\gcd(a,b)=1$. Let $d=\gcd(a+b,a^2+b^2)$. I want to prove that $d$ equals $1$ or $2$.
I get that $d\mid2ab$ but I can't find a linear combination that will give me some help to use the fact that $\gcd(a,b)=1$.
| Bezout based approach
Since $(a,b)=1$, there are $x,y$ so that $ax+by=1$. Then
$$
\begin{align}
1
&=(ax+by)^3\\
&=a^2\color{#C00000}{(ax^3+3bx^2y)}+b^2\color{#00A000}{(3axy^2+by^3)}
\end{align}
$$
Then, since
$$
(a^2+b^2)+(a+b)(a-b)=2a^2
$$
and
$$
(a^2+b^2)-(a+b)(a-b)=2b^2
$$
we have
$$
\begin{align}
&(a^2+b^2)\left[\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Proving that function with domain (-1,1) is injective. Function $g\colon (-1,1) \rightarrow \mathbb R$ is defined by $g(x)=\dfrac{x}{1-x^2}$. Prove that $g(x)$ is injective.
Work:
I shifted the equation so that it ends up like $\dfrac{x}{y}=\dfrac{1-x^2}{1-y^2}$. From here, I thought about using cases such as $x>y$ or... | Rewrite your function as follows $f(x) = \frac{x}{1-x^2} = \frac{1}{\frac{1}{x} - x}$
Suppose $f(x) = f(y)$. That is
$$ \frac{1}{\frac{1}{x} - x} = \frac{1}{\frac{1}{y} - y} \iff \frac{\frac{1}{y}-y}{\frac{1}{x} - x} = 1$$
We want to show $x = y$. Suppose not. Then we either have $x > y$ or $y > x$.
Suppose $y > x$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/758624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove for all $n \in N$, $\gcd(2n+1,9n+4)=1$ Question: Prove for all $n \in N$, $\gcd(2n+1,9n+4)=1$
Attempt: I want to use Euclid's Algorithm because it seemed to be easier than what my book was doing which was manually finding the linear combination.
Euclid's Algorithm states that we let $a,b \in N $. By applying the ... | There is also the systematic matrix approach:
$$
\begin{pmatrix} a \\ b \end{pmatrix}
=
\begin{pmatrix} 2 & 1 \\ 9 & 4 \end{pmatrix}
\begin{pmatrix} n \\ 1 \end{pmatrix}
\implies
\begin{pmatrix} n \\ 1 \end{pmatrix}
=
\begin{pmatrix} -4 & \hphantom-1 \\ \hphantom-9 & -2 \end{pmatrix}
\begin{pmatrix} a \\ b \end{pm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/758908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
A tough inequality problem with condition $a+b+c+abc=4$ If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality:
$$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$
I couldn't do anything, please help. However, from the some values I h... | We have by Holder's inequality:
$$\left(\sum_{cyc} \frac{a}{\sqrt{b+c}}\right)^2\left(\sum_{cyc}a(b+c) \right) \ge (a+b+c)^3$$
So it is sufficient to show that
$$2(a+b+c) \ge \sum_{cyc} a(b+c) \text{ or equivalently, } a+b+c \ge ab + bc + ca$$
Suppose $a+b+c < ab + bc + ca$. Then by Schur's inequality we have
$$\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/760759",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Finding the Asymptotic Curves of a Given Surface I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$.
I guess that what was meant by that statement is that surface $S$ can be locally parametrized by $$X(u,v) = \left( u, v, a \left( \fr... | Asymptotic lines for a Monge patch X[x,y] has a simple form. Primes on arc.
$ r\, x' ^2 + 2 \,s\, x'y' + t \, y' ^2 = 0 $ where $(r,t,s)$ are second partial derivatives.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/761350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Prove the matrix satisfies the equation $A^2 -4A-5I=0$ How to prove that
$$
A=\begin{bmatrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 1
\end{bmatrix}
$$
satisfies the equation $A^2 -4A-5I=0$?
| $$A^2 = \left(
\begin{array}{ccc}
9 & 8 & 8 \\
8 & 9 & 8 \\
8 & 8 & 9 \\
\end{array}
\right)$$
$$-4A = \left(
\begin{array}{ccc}
-4 & -8 & -8 \\
-8 & -4 & -8 \\
-8 & -8 & -4 \\
\end{array}
\right)$$
$$A^2 - 4A = \left(
\begin{array}{ccc}
5 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 5 \\
\end{array}
\right)$$
This is jus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/762051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
What mistake am I making trying to calculate the line integral $\oint_C3xy^2dx+8x^3dy$.
Evaluate the line integral
$$\oint_C3xy^2dx+8x^3dy$$
where $C$ is the boundary of the region between the circles $x^2+y^2=1$ and $x^2+y^2=64$ having positive orientation.
I actually used Green's theorem to find this. I know $r... | A direct integration around the two circles involves (by convention) following the larger circle counter-clockwise ("positive" direction) and then the smaller circle clockwise ("negative" direction). Applying polar coordinates, the integral becomes
$$ \int_0^{2 \pi} \ 3 \ (r \cos \theta) \ (r \sin \theta)^2 \ \ d(r \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/762366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent? Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent ?
I use it to compare with $1/n^2$, and then I used LHôpitals rule multiple times. Finally , I can solve it. However,I think we have other methods! somebody can help me?
| Note that if $$k\le\sqrt{n}< k+1$$ then, since the function $x\mapsto 3^x$ is increasing, we have $$3^k\le3^{\sqrt{n}}<3^{k+1},\;\;\text{and}$$ $$\frac{1}{3^k}\ge\frac{1}{3^{\sqrt{n}}}>\frac{1}{3^{k+1}}$$for every $n\in [k^2,(k+1)^2[$ then
\begin{align}
\sum_{n=1}^{N^2-1}{\frac{1}{3^{\sqrt{n}}}} & = \frac{1}{3^1}+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/762695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Logarithmic problem with 2 variables help How on earth do I solve? Any help will be much appreciated.
The value of $M$ is given by $M = a \log_{10}S + b$.
Note: Seismic moment measure the energy of the earthquake.
Using the following information, determine the values of $a$ and $b$ and hence find the seismi... | You have
\begin{align}
M=a\log_{10}S+b
\end{align}
Then, from the given values you have
\begin{align}
7=a\log_{10}(4.47\times10^{25})+b\tag1
\end{align}
and
\begin{align}
7.5=a\log_{10}(2\times10^{27})+b\tag2
\end{align}
Subtract $(1)$ from $(2)$ yield
\begin{align}
7.5-7&=a\log_{10}(2\times10^{27})+b-(a\log_{10}(4.47\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/764296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$ a+b+c+d=6 , a^2+b^2+c^2+d^2=12$ $\implies$ $ 36 \leq 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \leq48 $ Let $a , b, c, d$ be real numbers such that $$ a+b+c+d=6 \\ a^2+b^2+c^2+d^2=12$$ How do we prove that $$ 36 \space \leq\space 4(a^3+b^3+c^3+d^3)-(a^4+b^4+c^4+d^4) \space\leq48\space$$ ?
|
lemma1 : $$0\le a,b,c,d\le 3$$
proof: Use Cauchy-Schwarz inequality,we have
$$b^2+c^2+d^2\ge\dfrac{1}{3}(b+c+d)^2\Longrightarrow 12-a^2\ge \dfrac{(6-a)^2}{3}$$
so
$$\Longrightarrow 0\le a\le 3$$
lemma 2:
$$4a^3-a^4\ge 2a^2+4a-3,0\le a\le 3$$
proof:
$$\Longleftrightarrow a^4-4a^3+2a^2+4a-3\le 0$$
since
$$a^4-4a^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/767189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
How can I get the following recursive relation that explained? if $b(n)$ is the number of words created by the alphabet ${a,b,c}$ with $n$ length that each word has at least one $a$ character and after each $a$ there is no $c$ character write a recursive relation for $b(n)$.
I have tried calculating the words by mind a... | This is how I derived the recurrence.
Imagine the count $b(n)$ of good words to be split into two disjoint counts:
*
*$\color{blue}{x(n)}$: count of good words which end with $a$.
*$\color{green}{y(n)}$: count of good words which end with $b$ or $c$.
where a good word is defined as a word that meets our requireme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/767614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Whats the limit: $\lim\limits_{x \to 0} \frac{1-\sqrt[3]{\cos x}}{x(1-\cos\sqrt{x})}$ I am stuck at this one, i want to find it without using L'Hôpital's Rule.
$$\lim_{x \to 0} \frac{1-\sqrt[3]{\cos x}}{x(1-\cos\sqrt{x})}$$
| $1-\cos \alpha=2\sin^2\frac{\alpha}2$, $1-\cos \alpha\sim \frac {\alpha^2} 2(\alpha\to0)$, so
\begin{equation}
\begin{split}\lim_{x \to 0} \frac{1-\sqrt[3]{\cos x}}{x(1-\cos\sqrt{x})} &=\lim_{x \to 0} \frac{1-\cos x}{x(1-\cos\sqrt{x})(1+\sqrt[3]{\cos x}+\sqrt[3]{\cos^2 x})}\\&=\frac13 \lim_{x \to 0} \frac{1-\cos x}{x(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/767885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How would I solve this system of differential equations? Find a general solution of
$$
x' = \begin{pmatrix}
11 & -25 \\
4 & -9
\end{pmatrix} x + \begin{pmatrix}
e^{-t} \\
0
\end{pmatrix} $$
I found the general solution of the homogeneous system to be
$$ c_1e^t \begin{pmatrix}
5 \\
2
\end{pmatrix} +... | I agree with the general solution you found.
See my post here for details of this process, resolve an non-homogeneous differential system.
We have:
$$\phi(t) = e^t \left(
\begin{array}{cc}
5 & 5 t+3 \\
2 & 2 t+1 \\
\end{array}
\right)$$
$$\phi^{-1}(t) = \left(
\begin{array}{cc}
-e^{-t} (2 t+1) & e^{-t} (5 t+3) \\
2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/768571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How Do I Integrate? $\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}$ How do I integrate this one? $$\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}\,dx$$
Is my answer correct: $$-3\ln\left \| x+2 \right \|+\ln\left \| x \right \|+\frac{4}{x}+C$$
| Partial fraction decomposition.
$$\dfrac{-2x^2 + 6x +8}{x^2(x + 2)} = \dfrac{A}{x}+ \dfrac{B}{x^2} + \dfrac{C}{x+2}$$
Solve for $A, B, C$.
Your almost there with your answer, but we need $-\frac 4x$, and $\ln |x| - 3\ln|x+2| = \ln\left|\dfrac{x}{(x+2)^3}\right|$, if you want to simplify further.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/768834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Testing for convergence of a series Test the following series for convergence
$$\sum_{k=1}^\infty \sin k \sin \frac{1}{k}$$.
I am just wondering if the following method is ok:
$$\sum_{k=1}^\infty \sin k \sin \frac{1}{k} = \sum_{k=1}^\infty \sin (\frac{1}{k} - (\frac{1}{k} -k)) \sin \frac{1}{k} \\ \leq \sum_{k=1}^\infty... | Use the fact that $\sum \frac{ \sin (n)}{n} $ converges. Hence, by limit comparison test,
$$ \frac{ \sin (n) \sin ( \frac{1}{n} ) }{\frac{ \sin (n)}{n}} = \frac{ \sin( \frac{1}{n} )}{\frac{1}{n}} \to 1 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/769867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Power Series Solution to Differential Equation The equation is
$$y'' - xy' + y = 0$$
So far I have the recurrence relation -
$$a_{n+2} = \dfrac{(n-1)a_n}{(n+1)(n+2)} $$
From this -
$a_2 = \dfrac{-a_0}{2!}$
$a_3 = 0$
$a_4 = \dfrac{-a_0}{4!}$
$a_5 = 0$
$a_6 = \dfrac{-3a_0}{6!}$
and so on..
The question asks for the fir... | The differential equation
\begin{align}
y^{''} - x y^{'} + y = 0
\end{align}
can be solved via a power series of the form
\begin{align}
y(x) = \sum_{k=0}^{\infty} a_{n} x^{n} = a_{0} + a_{1} x + a_{2} x^{2} + \cdots .
\end{align}
It is fairly evident that
\begin{align}
\sum_{k=0}^{\infty} k(k-1) a_{k} x^{k} = \sum_{k=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/771981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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what are the integral sides possible? Let a triangle have sides $a$,$b$,$c$ and $c$ is the greatest side , triangle ABC is obtuse in nature having integral sides. Find the smallest perimeter possible
Given A=2B where A and B are angles of respective vertices
| Basic conditions for enumeration
A brute force approach could enumerate triangles with integral edge lengths in order of increasing perimeter. In other words, examine partitions of increasingly larger numbers into three summands, taking the triangle inequality into account. What we'd need is some easy approach to detec... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve $\cos x+8\sin x-7=0$
Solve $\cos x+8\sin x-7=0$
My attempt:
\begin{align}
&8\sin x=7-\cos x\\
&\implies 8\cdot \left(2\sin \frac{x}{2}\cos \frac{x}{2}\right)=7-\cos x\\
&\implies 16\sin \frac{x}{2}\cos \frac{x}{2}=7-1+2\sin ^2\frac{x}{2}\\
&\implies 16\sin \frac{x}{2}\cos \frac{x}{2}=6+2\sin^2 \frac{x}{2}\\
&\i... | Divide either sides by $\cos^2\frac x2$ $$\sin^2\frac x2-8\sin\frac x2\cos\frac x2+3=0$$ to get $$\tan^2\frac x2-8\tan\frac x2+3\left(1+\tan^2\frac x2\right)=0$$ which on rearrangement is a Quadratic Equation in $\displaystyle\tan\frac x2$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Integral $\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}$ Hi I am trying to show$$
I:=\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}.
$$
Thank you.
What a desirable thing to want to prove! It is a wo... | I will solve the general form
\begin{align}\int^\infty_0\frac{x}{x^2+1}\frac{dx}{\sinh(ax)}&=\int^\infty_0\frac{x}{x^2+a^2}\frac{dx}{\sinh(x)}
\\&=\int^\infty_0 \int^\infty_0e^{-at} \frac{\sin(xt)}{\sinh(x)}\,dt \, dx\\&=\int^\infty_0 e^{-at}\int^\infty_0 \frac{\sin(xt)}{\sinh(x)} \, dx\,dt
\\&=\frac{\pi}{2} \int^\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/776679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 4,
"answer_id": 3
} |
Elegant proof of $\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$? Let $a, b > 0$ satisfy $a^2-4b^2 \geq 0$. Then:
$$\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$$
One way to calculate this is by computing the residues at the poles in the upper h... | It's time to return the favor from here Ruben. (>‿◠)✌
First, we will prove
$$\int_{-\infty}^\infty \frac{dx}{\beta^4x^4+2\beta^2\cosh(2\alpha)\,x^2+1}=\frac{\pi}{2\beta\cosh\alpha}$$
Note that
$$\frac{1}{\beta^4x^4+2\beta^2\cosh(2\alpha)\,x^2+1}=\frac{1}{e^{2\alpha}-e^{-2\alpha}}\left[\frac{1}{\beta^2 x^2+e^{-2\alpha}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/776812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 0
} |
jacobian times its inverse - should be identity Here's an easy one.
A Jacobian is $\frac{dx^i}{dy^j}$.
The inverse is $\frac{dy^j}{dx^k}$. So, in tensor notation,
$\frac{dx^i}{dy^j} \frac{dy^j}{dx^k} = \frac{dx^i}{dx^k} = \delta^i_k$
Now I'll try to this as in matrix form, in two dimensions:
$\left[
\begin{arra... | In two dimensions, we have functions $f_1$ and $f_2$ which are both functions of $x_1$ and $x_2$. The Jacobian matrix is given by,
$$J=\left(
\begin{matrix}
\partial_1 f_1 & \partial_2 f_1 \\
\partial_1 f_2 & \partial_2 f_2
\end{matrix}
\right)$$
where $\partial_n := \partial/\partial x_n$. Computing the inverse of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/778149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A question on egyptian fractions An Egyptian fraction is the sum of distinct unit fractions. Are there any 2000 egyptian fractions that their sum is 1?
| Yes!
Because for any $k\geqslant 3$ we can write $1$ as sum of $k$ different fractions $\dfrac{1}{n}$. $\dfrac{1}{n}=\dfrac{1}{n+1}+\dfrac{1}{n(n+1)}$.
Apllying this identity we get: $1=\dfrac{1}{2}+\dfrac{1}{2}=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{6}=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{7}+\dfrac{1}{42}=\dots$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/778227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
To find relatively prime ordered pairs of positive integers $(a,b)$ such that $ \dfrac ab +\dfrac {14b}{9a}$ is an integer How many ordered pairs $(a,b)$ of positive integers are there such that g.c.d.$(a,b)=1$ , and
$ \dfrac ab +\dfrac {14b}{9a}$ is an integer ?
| I first find out what the values of a/b are which make the expression an integer, then check which of the solutions are rational numbers. Let t = a/b, then t + 14/(9t) should be an integer.
Solve t + 14/(9t) = n: t^2 + 14/9 = nt, t^2 - nt + n^2/4 = n^2/4 - 14/9.
t - n/2 = +/- sqrt (n^2/4 - 14/9).
t should be rational... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Are there nice ways to solve $(2+x)^{0.25}-(2-x)^{0.25}=1$ Are there nice and elegant ways to solve this equation?
$(2+x)^{0.25}-(2-x)^{0.25}=1$
Thanks.
| Let $A=\sqrt{2+x}$ and $B=\sqrt{2-x}$. See if you can show the following two equations hold:
$$A^2+B^2=4$$
$$2A+2AB+2B=5$$
Adding these two equations together, and using the quadratic formula you should be able to show that $A+B=\sqrt{10}-1$. Then, plugging this into the second equation, and using the fact that $AB=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Remarkable integral: $\int_0^{\infty} x \left(1 - \frac{\sinh x}{\cosh x-\sqrt 3/2} \right) \mathrm dx= -\frac{13 \pi ^2}{72}$? Numerical evidence suggests that
$$\int_0^{\infty} x \left(1 - \frac{\sinh x}{\cosh x-\sqrt 3/2} \right) \mathrm dx= -\frac{13 \pi ^2}{72}$$
How can we prove this? I could not find a nice con... | The following is another approach that also involves differentiating under the integral sign.
Let $$I(\theta) = \int_{0}^{\infty} x \left(1- \frac{\sinh x}{\cosh x - \cos \theta} \right) \, \mathrm dx \, , \quad 0 < \theta < \pi. $$
Then $$I'(\theta) = \sin \theta \int_{0}^{\infty} \frac{x \sinh x}{\left(\cosh x - \cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
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Evaluate $\frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\cdots$ Evaluate
$$
\frac{2}{4}\frac{2+\sqrt{2}}{4}\frac{2+\sqrt{2+\sqrt{2}}}{4}\frac{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}{4}\cdots .
$$
First, it is clear that terms tend to $1$.
It seems that the infinity product is not 0. This is related to the post Se... | If we set $a_0=1/2$ and define
$$
a_k=\frac{1+\sqrt{a_{k-1}}}{2}\tag{1}
$$
then the product sought is
$$
\prod_{k=0}^\infty\ a_k\tag{2}
$$
Since
$$
\cos(2x)=2\cos^2(x)-1\tag{3}
$$
we have that
$$
a_k=\cos^2(2^{-k}x_0)\tag{4}
$$
satisfies $(1)$ with $x_0=\frac\pi4$. Then
$$
\sin(2x)=2\sin(x)\cos(x)\tag{5}
$$
implies by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/782156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Proving a tough geometrical inequality, with equality in equilateral triangles. For any triangle with sides-lengths $a$, $b$ and $c$ prove or disprove (1) and (2) :
*
*$$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$
*Equality in (1) holds if and only if the triangle is equilateral.
Playing w... | Let $a=y+z$, $b=x+z$ and $c=x+y$.
Hence, By C-S we obtain:
$$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}=\sum_{cyc}\frac{a^2}{(a+b)^2-c^2+a^2}=$$
$$=\sum_{cyc}\frac{(y+z)^2}{y^2+5z^2+6yz+4zx}=\sum_{cyc}\frac{(y+z)^2(y+x)^2}{(y+x)^2(y^2+5z^2+6yz+4zx)}\geq$$
$$\geq\frac{\left(\sum\limits_{cyc}(y+z)(y+x)\right)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/783808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Find the value of the integral Find the value of the following integral conatining a term with natural logarithm$$\int_0^1 (1-y) \ln\left(\frac{2+\sqrt{1-y}}{2-\sqrt{1-y}}\right)\, dy.$$
| Rewrite $\displaystyle\int_0^1 (1-y)\ln\left(\frac{2+\sqrt{1-y}}{2-\sqrt{1-y}}\right)\ dy$ as
$$
\int_0^1 (1-y) \ln\left(2+\sqrt{1-y}\right)\ dy-\int_0^1 (1-y) \ln\left(2-\sqrt{1-y}\right)\ dy
$$
then let $x=2+\sqrt{1-y}$ and $z=2-\sqrt{1-y}$. The integral becomes
$$
\int_0^1 (1-y) \ln\left(2+\sqrt{1-y}\right)\ dy=2\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/787686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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How prove this limit $\lim_{n\to\infty}\sum_{k=n}^{9n-1}\frac{1}{\sqrt{k^2+1}+k}=\ln{3}$
show that: this limit
$$\lim_{n\to\infty}\sum_{k=n}^{9n-1}\dfrac{1}{\sqrt{k^2+1}+k}=\ln{3}$$
my idea: since
$$\dfrac{1}{\sqrt{k^2+1}+k}=\sqrt{k^2+1}-k$$
then I can't.Thank you
| Notice $\frac{1}{\sqrt{k^2+1}+k}$ is monotonic decreasing, we have
$$\sum_{k=n}^{9n-1}\frac{1}{\sqrt{k^2+1}+k} \ge
\int_n^{9n}\frac{dx}{\sqrt{x^2+1}+x} \ge \sum_{k=n+1}^{9n}\frac{1}{\sqrt{k^2+1}+k}\\
$$
This implies
$$\left| \sum_{k=n}^{9n-1}\frac{1}{\sqrt{k^2+1}+k} -
\int_n^{9n}\frac{dx}{\sqrt{x^2+1}+x} \right|
\le \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/788719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
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Stuck with integral Having this:
$\int x\sqrt{1-x^2}dx$
Substitution:
$t = 1-x^2$
$dt = -2xdx => dx=\frac{-2x}{dt}$
So:
$$\int x\sqrt{1-x^2}dx = -\int x t^\frac{1}{2}\frac{2x}{dt} = -\int \frac{2x^2 t^\frac{1}{2}}{dt} = $$
...but here I stuck... I've tried $-4x\frac{t^\frac{-1}{2}}{\frac{-1}{2}} + c$ but it doesn't mat... | Use the fact that $\frac{d}{dx}((1-x^2)^\frac{3}{2})=-3x(1-x^2)^\frac{1}{2}$
(This is just using the rule $\frac{d}{dx}f(x)^n=f'(x)f(x)^{n-1}$)
And we have:
$\int x\sqrt{1-x^2}dx=(-1/3)\int 3x\sqrt{1-x^2}dx=(-1/3)\int \frac{d}{dx}((1-x^2)^\frac{3}{2})dx$
$=\int 1d((1-x^2)^\frac{3}{2})=-(1/3)(1-x^2)^\frac{3}{2}+c$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/791992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Find the linear-to-linear function whose graph passes through the given three points Find the linear-to-linear function whose graph passes through the points
$(1, 1)$, $(4, 2)$ and $(30, 3)$.
So by using the
$$f(x)=\frac{ax +b}{x+d}$$
I got my final answer to be
$$f(x)=\frac{\frac{75}{23}x + \frac{64}{23}}{x + \frac... | So we know that $f(1) = 1$, $f(4) = 2$, and $f(30) = 3$.
Since $f(x)$ is a linear to linear function, we know that:
$f(x) = (ax + b)/(x + c)$
Substituting, we have:
$f(1) = (a + b)/(1 + c)$ = 1, or $a + b = 1 + c$
$f(4) = (4a + b)/(4 + c)$ = 2, or $4a + b = 8 +2c$
$f(30) = (30a + b)/(30 + c)$ = 3, or $30a + b = 90 +3c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/795405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does $\frac{4}{2} = \frac{2}{1}$? I take for granted that $\frac{4}{2} = \frac{2}{1}$.
Today, I thought about why it must be the case. My best answers amounted to $\frac{4}{2}=2$ and $\frac{2}{1}=2$; therefore $\frac{4}{2}=\frac{2}{1}$. However, that explanation seems circular:
*
*one can express $2$ as $\frac... | The set $\mathbb{Q}$ is a group then for all $x\in \mathbb{Q}$, there is a only Inverse element such that $x+(-x)=0$. In this case note that
$$\frac{4}{2}-\frac{2}{1}=0$$
then you conclude that $$\frac{4}{2}=\frac{2}{1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/798687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
Finding generators for an ideal of $\Bbb{Z}[x]$ We know that $\Bbb{Z}$ is Noetherian. Hence, we can conclude that $\Bbb{Z}[x]$ is Noetherian, too. Consider the ideal generated by $\langle 2x^2+2,3x^3+3,5x^5+5,…,px^p+p,…\rangle$ for all prime natural numbers $p$. How can I determine a finite number of elements which gen... | Hint Use the Euclidian Algorithm in $\mathbb{Q}[X]$, then multiply whatever what you get by the common denominator. If needed.
Edit By the extended EA we have
$$3x^3+3=\frac{3}{2}x (2x^2+2)+3-3x$$
$$2x^2+2=-\frac{2}{3}(-3x+3)+4$$
Therefore
$$4=2x^2+2+\frac{2}{3}(3x^3+3-\frac{3}{2}x (2x^2+2))\\
=(2x^2+2)(x-1)+\frac{2}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/802806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Integral $\int_0^\infty \log \frac{1+x^3}{x^3} \frac{x \,dx}{1+x^3}=\frac{\pi}{\sqrt 3}\log 3-\frac{\pi^2}{9}$ I am trying to prove this interesting integral
$$
I:=\int_0^\infty \log \frac{1+x^3}{x^3} \frac{x \,dx}{1+x^3}=\frac{\pi}{\sqrt 3}\log 3-\frac{\pi^2}{9}.
$$
I tried using $y=1+x^3$ but that didn't help.
We c... | Define
$$
I(a)=\int_0^\infty \log \frac{a+x^3}{x^3} \frac{x \,dx}{1+x^3}.$$
Then $I(0)=0$ and
\begin{eqnarray}
I'(a)&=&\int_0^\infty\frac{x}{(a+x^3)(1+x^3)}dx\\
&=&\frac13\int_0^\infty\frac{1}{x^{1/3}(a+x)(1+x)}dx\\
&=&\frac{1}{3(1-a)}\left(\int_0^\infty\frac{1}{x^{1/3}(a+x)}dx-\int_0^\infty\frac{1}{x^{1/3}(1+x)}dx\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
How to derive $\frac{d}{dx}\left(x+1\right)^{\sin\left(x\right)}$ I need help to find derivative of: $\frac{d}{dx}(x+1)^{\sin x}$
i tried to do something like this..
$$(x+1)^{\sin x}\cdot \ln\left(x+1\right)=\sin x(x+1)^{\sin\left(x\right)-1}\cdot \ln(x+1)-(x+1)^{\sin(x)}\cdot \frac{1}{x+1}\:=\sin x(x+1)^{\sin\left(x\... | You have:
$$f(x) = (x+1)^{\sin{x}},$$ so $\ln{f} = \sin{x} \, \ln{(x+1)}$. Now we have:
$$ \frac{d}{dx} \ln {f} = \frac{1}{f} \frac{df}{dx} = \frac{d}{dx}\left[ \sin{x} \, \ln{(x+1)} \right] . $$
You can now solve for $f'$ once you expand the LHS of the equation.
Cheers!
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Does this sequence of polynomials have a closed form? Consider the following sequence of polynomials in the variable $d$, which I encountered during a calculation:
$$
1 + 3 d^2
\\
1 + 10 d^2 + 5 d^4
\\
1 + 21 d^2 + 35 d^4 + 7 d^6
\\
1 + 36 d^2 + 126 d^4 + 84 d^6 + 9 d^8
\\
1 + 55 d^2 + 330 d^4 + 462 d^6 + 165 d^8 + 11 ... | It looks like $\dfrac{(1+d)^{2k+1}+(1-d)^{2k+1}}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/805067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Is my proof correct regarding the non primality of $2\cdot 17^a +1$? Today I need your help to know if the proof I have provided below is correct or not. I want to prove that there is no prime of the form $2\cdot 17^a+1$ where $a\in \mathbb N$.
Now, first of all, I tried to get some initial information about $a$ and us... | No, your second computation and third computation rule out exactly the same candidates, all the $a$ congruent to $1$ mod $4$. You have never adressed the case where $a \equiv 3 \pmod 4$
There is absolutely no hope of doing a similar computation ruling out all the $a$ congruent to $3$ mod $4$, because the case $a = 47$ ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Calculating limits using l'Hôpital's rule. After a long page of solving limits using l'Hôpital's rule
only those 2 left that i cant manage to solve
$$\lim\limits_{x\to0}{\sqrt {\cos x} - \sqrt[3]{\cos x}\over \sin^2 x }$$
$$\lim\limits_{x\to\ {pi\over 2}}{\tan 3x - 3\over \tan x - 3 }$$
Thanks in advance for any help ... | $$\lim\limits_{x\to0}{\sqrt {\cos x} - \sqrt[3]{\cos x}\over \sin^2 x }=\lim\limits_{x\to0}{-\frac{1}{2}\sin x(\cos x)^{-\frac{1}{2}} + \frac{1}{3}\sin x (\cos x)^{-\frac{1}{3}}\over 2\sin x \cos x }=\lim\limits_{x\to0}{-\frac{1}{2}(\cos x)^{-\frac{1}{2}} + \frac{1}{3} (\cos x)^{-\frac{1}{3}}\over 2 \cos x }=-\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How find this sum $ \frac{1}{1999}\binom{1999}{0}-\frac{1}{1998}\binom{1998}{1}+\cdots-\frac{1}{1000}\binom{1000}{999}$
prove or disprove :
$$S=\dfrac{1}{1999}\binom{1999}{0}-\dfrac{1}{1998}\binom{1998}{1}+\dfrac{1}{1997}\binom{1997}{2}-\dfrac{1}{1996}\binom{1996}{3}+\cdots-\dfrac{1}{1000}\binom{1000}{999}=\dfrac{1}{1... | The Chebyshev polynomials of the first kind are given by
\begin{align}
T_{n}(x) &= \frac{n}{2} \sum_{k=0}^{[n/2]} \frac{(-1)^{k}}{n-k} \binom{n-k}{k} (2x)^{n-2k} \\
&= \frac{1}{2} \left[ (x - \sqrt{x^{2}-1})^{n} + (x - \sqrt{x^{2}-1})^{n} \right].
\end{align}
When $x=1/2$ it is seen that
\begin{align}
\sum_{k=0}^{[n/2]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 2,
"answer_id": 0
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How prove $\left(\frac{b+c}{a}+2\right)^2+\left(\frac{c}{b}+2\right)^2+\left(\frac{c}{a+b}-1\right)^2\ge 5$ Let $a,b,c\in R$ and $ab\neq 0,a+b\neq 0$. Find the minimum of:
$$\left(\dfrac{b+c}{a}+2\right)^2+\left(\dfrac{c}{b}+2\right)^2+\left(\dfrac{c}{a+b}-1\right)^2\ge 5$$
if and only if $$a=b=1,c=-2$$
My idea: Since ... | Thank you ,@Calvin Lin,
I have solve this problem ,let $$F(a,b,c)=\left(\dfrac{b+c}{a}+2\right)^2+\left(\dfrac{c}{b}+2\right)^2+\left(\dfrac{c}{a+b}-1\right)^2$$
note
$$F(a,b,c)-5=\dfrac{\left((a^2+ab+b^2)c+b(a+b)(2a+b)\right)^2}{a^2b^2(a+b)^2}\ge 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/807404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Non-linear Second Order Differential equation I want to ask for a hint in solving the following ODE,
$y'' (1+x^2) + y' *(x) = C$,
where C is a constant. I've tried a couple of ways to manipulate this ODE; such as letting v = y'
$v' (1+x^2) + v *(x) = C$
but I can't see a way to solve the equation for y
| $$
\left(1+x^2\right)v' + xv = C
$$
we can obtain
$$
v\sqrt{1+x^2} = C\int\frac{1}{\sqrt{1+x^2}}dx + \lambda_{1}
$$
or
$$
\frac{dy}{dx} = \frac{C}{\sqrt{1+x^2}}\int\frac{1}{\sqrt{1+x^2}}dx + \frac{\lambda_{1}}{\sqrt{1+x^2}} = \frac{1}{\sqrt{1+x^2}}\left[C\sinh^{-1}(x) + \lambda_1\right]
$$
so
$$
y = \lambda_1\int \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/808711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to prove: prove $\frac{1+\tan^2\theta}{1+\cot^2\theta} = \tan^2\theta$ I need to prove that $\frac{1+\tan^2\theta}{1+\cot^2\theta}= \tan^2\theta.$
I know that $1+\tan^2\theta=\sec^2\theta$ and that $1+\cot^2\theta=\csc^2\theta$, making it now $$\frac{\sec^2\theta}{\csc^2\theta,}$$ but I don't know how to get it do... | Draw a right triangle A and label the sides appropriately as OPP, ADJ, and HYP with angle $\theta$ opposite side OPP, and HYP opposite the right angle. Next generate similar triangle B by multiplying each side of triangle A by $\displaystyle\frac{HYP}{OPP\times ADJ }$.
Triangle B has a hypotenuse $\displaystyle HYP\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 9,
"answer_id": 3
} |
proving $\tan^{-1} \left( \frac{x}{1-x^2} \right) = \tan^{-1}x + \tan^{-1}(x^3)$ Two related questions, one easy, one just a bit harder:
1) Prove the identity
$$
\tan^{-1} \left( \frac{x}{1-x^2} \right) =
\tan^{-1}x + \tan^{-1}(x^3)
$$
2) Now try to find a geometric or trigonometric proof of that same geometry, without... | The proof can be established starting from the angle addition identity $$\sin(x+y) = \sin x \cos y + \cos x \sin y.$$ Then with $x = \frac{\pi}{2} - a$, $y = -b$, we get $$\cos(a+b) = \sin(\tfrac{\pi}{2} - a - b) = \cos a \cos b - \sin a \sin b.$$ We then find $$\tan(x+y) = \frac{\sin(x+y)}{\cos(x+y)} = \frac{(\sin x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/812123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove that $2002^{2001}$ and $2002^{2001}+2^{2001}$ have the same number of digits. I have an arithmetic exercise as follows: Prove that $2002^{2001}$ and $2002^{2001}+2^{2001}$ have a same number of digits. It seems easy but I don't know how to do. I can do it if $2002$ is replaced by $2000$. Any hints are appreciated... | $\ln (a + b) = \ln (a(1 + \dfrac{b}{a})) = \ln a + \ln (1 + \dfrac{b}{a})$.
The number of digits in the base-10 numeral for integer $n > 0$ is $\lceil \log_{10}n \rceil$. $\log_{10} n = \dfrac{\ln n}{\ln 10}$
Let $a = 2002^{2001}$ and $b = 2^{2001}$
$\lceil \log_{10} a \rceil = \left\lceil \dfrac{\ln a}{\ln 10} \right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/813773",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
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Applications of the identity $ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$ I am reading Euclid's elements
I found the algebraic identity
$ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$
I ponder on usage of this identity for $2$ hours.
but I can't click anything.
$a^2 + b^... | This identity can be rewritten as $$(a-b)^2=(a+b)^2-4ab$$ and, is the secret behind the discriminant of a quadratic equation. Let me explain it. Suppose $Ax^2+Bx+C=0$ is a quadratic equation with real coefficients, then its roots must occur as complex conjugate pairs. If we have two complex roots $a, b$ for this equati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 5
} |
Finding all possible values of $x^4+y^4+z^4$ Given real numbers $x,y,z$ satisfying $x+y+z=0$ and
$$ \frac{x^4}{2x^2+yz}+\frac{y^4}{2y^2+zx}+\frac{z^4}{2z^2+xy}=1$$
Find all possible values of $x^4+y^4+z^4$ with proof.
My attempt :
Putting $x=-y-z$ and doing subsequent calculations we get something like $$\displaystyle... | this is a simple approach for $x^2+xy+y^2=1$, The op has a good start anyway, from his last step:
note:$\dfrac{x^4}{(z-x)(y-x)}=\dfrac{x^4}{y-z}\left(\dfrac{1}{z-x}-\dfrac{1}{y-x}\right)$
$\sum_{cyc}\dfrac{x^4}{(z-x)(y-x)}=\dfrac{x^4}{(z-x)(y-z)}-\dfrac{x^4}{(y-z)(y-x)}+\dfrac{y^4}{(z-y)(x-y)}+\dfrac{z^4}{(x-z)(y-z)}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Solve a system of linear equations $\newcommand{\Sp}{\phantom{0}}$There is a system of linear equations:
\begin{alignat*}{4}
&x - &&y - 2&&z = &&1, \\
2&x + 3&&y - &&z =-&&2.
\end{alignat*}
I create the matrix of the system:
$$
\left[\begin{array}{rrr|r}
1 & -1 & -2 & 1 \\
2 & 3 & -1 & -2
\end{array}\rig... | I would apply two more row operations and present a solution with a free variable:
$$
\begin{align*}
& \left[\begin{array}{rrr|r}
1 & -1 & -2 & 1 \\
2 & 3 & -1 & -2
\end{array}\right] R_2+(-2)R_1 \rightarrow R_2 \\
\equiv & \left[\begin{array}{rrr|r}
1 & -1 & -2 & 1 \\
0 & 5 & 3 & -4
\end{array}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/817183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Equation of a tangent line to a curve Equation of the curve is $y=(x+9)/(x+5)$, we are looking for the tangent line to that curve that also goes through $O(0,0)$. Answer given is $x+25y=0$ which I found to be true for $A(-15, 3/5)$ being part of the line and the curve. Question is, how we got to that answer. After diff... | This doesn't always work, but when it does, it means that you can find the equation of the tangent line without using calculus.
A line through the origin must have the form $y_{_L} = mx$. When does such a line intersect the graph of the equation $y_{_H}=\dfrac{x+9}{x+5}$?
\begin{align}
y_{_L} &= y_{_H} \\
mx &= \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculate the number of different words, where $0$ appears an even number of times. Let the set of words with an even length $n$ from the alphabet: $\{ 0,1,2\}$.
Calculate the number of different words, where $0$ appears an even number of times.
For example, for $n=6$ , the words $121212,001212,000000$ are allowed,but ... | Another way is by using exponential generating function:
Since $0$ appears even number of times and $1,2$ has no restriction, the egf is:
\begin{align*}
G(x) &= \left(1+\frac{x^2}{2}+\frac{x^4}{4!}+\cdots\right)\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\right)^2 \\
&= \left(\frac{e^x+e^{-x}}{2}\right)e^{2x} \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$
My approach :
I used $\sin A +\sin B = 2\sin(A+B)/2\times\cos(A-B)/2 $
$\Rightarrow \sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7} = 2\sin(3\pi)/7\times\co... | Here's my approach, it only use basic trigonometry identities (i.e, it requires no complex numbers), so it's quite lengthy. I would be glad if you guys can help me shorten it a bit.
*
*Use Product to Sum Formula to prove that:
$\displaystyle \begin{align*}& \ \cos\left(\frac{2\pi}{7} \right) \cos\left(\frac{4\pi}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 1
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The 3 Integral $\int_0^\infty {x\,{\rm d}x\over \sqrt[3]{\,\left(e^{3x}-1\right)^2\,}}=\frac{\pi}{3\sqrt 3}\big(\log 3-\frac{\pi}{3\sqrt 3} \big)$ Hi I am trying evaluate this integral and obtain the closed form:$$
I:=\int_0^\infty \frac{x\,dx}{\sqrt[\large 3]{(e^{3x}-1)^2}}=\frac{\pi}{3\sqrt 3}\left(\log 3-\frac{\pi}{... | Substitute $e^{-x}=t$, then the integral can be written as:
\begin{align*}
\int_0^\infty \frac{x\,dx}{\sqrt[\large 3]{(e^{3x}-1)^2}}\, dx &= -\int_{0}^{1} \, \frac{t\, \log{t}}{\left(1-t^3\right)^{2/3}}\, dt\tag 1
\end{align*}
Consider:
\begin{align*}
I(a) &= \int_{0}^{1} \, \frac{t^{a+1}}{\left(1-t^3\right)^{2/3}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 2,
"answer_id": 0
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How to integrate $\int \frac{1}{\sin^4x + \cos^4 x} \,dx$?
How to integrate
$$\int \frac{1}{\sin^4x + \cos^4 x} \,dx$$
I tried the following approach:
$$\int \frac{1}{\sin^4x + \cos^4 x} \,dx = \int \frac{1}{\sin^4x + (1-\sin^2x)^2} \,dx = \int \frac{1}{\sin^4x + 1- 2\sin^2x + \sin^4x} \,dx \\
= \frac{1}{2}\int \... | Simplify the denominator in the following way:
$$\sin^4x+\cos^4x=(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x=1-\frac{\sin^2(2x)}{2}=\frac{1+\cos^2(2x)}{2}=\frac{2+\tan^2(2x)}{2\sec^2(2x)}$$
Hence, the integral you are dealing with is:
$$\int \frac{2\sec^2(2x)}{2+\tan^2(2x)}\,dx$$
I guess the next step is pretty obvious now. ;)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 2
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Inverting $a+b\sqrt{2}$ in the field $\Bbb Q(\sqrt{2})$ I have been reading through my notes and I came across this example and I found it hard to understand so I need some help in explaining how the inverse of this is found.
The set $\mathbb Q(\sqrt 2)=\{a+b\sqrt2$: $a,b$ in $\mathbb Q \}$ is a field. The inverse of a... | Consider an arbitrary element $a + b \sqrt{2} \in \mathbb{Q}[\sqrt{2}]$. From our experience with real numbers, we know that its inverse will be $\displaystyle \frac{1}{a + b\sqrt{2}}$. However, this is not in the "standard" form for elements in $\mathbb{Q}[\sqrt{2}] = \{x + y\sqrt{2} \ | \ x, y \in \mathbb{Q}\}$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
} |
Integrate : $\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}dx$
$$\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}\ dx$$
My approach :
Dividing the denominator by $\cos^2x$ we get $\dfrac{x^2\sec^2x }{(x -\tan x)(x\tan x +1)}$ then
$$\int \frac{x^2\sec^2x}{x^2\tan x -x\tan^2x+x-\tan x}\ dx=\int \frac{x^2(1+\t... | $\bf{My\; Solution::\; }$Let $\displaystyle I = \int\frac{x^2}{(x\sin x+\cos x)\cdot (x\cos x-\sin x)}dx$
Let $x=\tan \theta\;,$ Then $dx = \sec^2 \theta d\theta.\;\; $ Then $$I = \displaystyle \int \frac{\tan^2 \theta\cdot \sec^2 \theta }{(\tan \theta\cdot \sin(\tan \theta)+\cos(\tan \theta) )\times (\tan \theta\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 5,
"answer_id": 1
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Prove that if $2^{p}-1$ is prime then $n=2^{p-1}(2^p-1)$ is a perfect number Prove that if $2^{p}-1$ is prime then $$n=2^{p-1}(2^p-1)$$ is a perfect number
here is what i did:
We need to prove the $\sigma(n)=n$
so $\sigma(n)=\sigma(2^{p-1})\sigma(2^p-1)$
since $2^{p}-1$ is a prime thus $\sigma(2^p-1)=2^p$
since $2$ is ... | Let $p \geq 2$ and that $d=2^p-1$ is a prime.
Then,the divisors of $2^{p-1} d$ are these:
$$1,2, \dots, 2^{p-1},d,2d, \dots , 2^{p-1} d$$
Therefore,
$$\sigma(n)=1+2+ \dots+ 2^{p-1}+d+2d+ \dots+ 2^{p-1} d=(1+2+ \dots+ 2^{p-1})(1+d) \\ =(2^p-1)2^p=2n$$
So,we conclude that $n$ is a perfect number.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/822215",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Expand $(x-7)^2 + x^2 = (x+2)^2$ algebraically $(x-7)^2 + x^2 = (x+2)^2$
$(x-7)(x-7) + x^2 = (x+2)(x+2)$
$x^2 -7x -7x + 49 + x^2 = x^2 + 4x + 4$
$x^2 + 18x - 45 = x^2 + x^2$
From that point on, everything I do is incorrect. I don't know what to do with the three $x^2$.
| Expand the lhs and you get $lhs=2x^2-14x+49$. Expand the rhs and get $rhs=x^2+4x+4$.So $$lha-rhs=(2x^2-14x+49)-(x^2+4x+4)=x^2-18x+45$$ What you wrote is correct but you did not finish. You ended with $$x^2 + 18x - 45 = x^2 + x^2$$ so substract the last lhs from the last rhs of your post and you are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/822717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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If $\alpha = \frac{2\pi}{7}$ then the find the value of $\tan\alpha .\tan2\alpha +\tan2\alpha \tan4\alpha +\tan4\alpha \tan\alpha.$ If $\alpha = \frac{2\pi}{7}$ then the find the value of $\tan\alpha .\tan2\alpha +\tan2\alpha \tan4\alpha +\tan4\alpha \tan\alpha$
My 1st approach :
$\tan(\alpha +2\alpha +4\alpha) = \f... | We can usually solve these kind of problems by building an equation for which the the given trigonometric values are the roots. (See S.L.Loney's book)
It's as much time consuming as doing it by expanding and simplifying the angle sums, but we get several other answer's easily once the equation is in place.
Here is an o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
What is the sum of this? $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$ I'm in trouble with this homework.
Find the sum of the series $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$, where the terms are the inverse of the positive integers who... | $$\left(1+\frac{1}{2}+\frac{1}{4}+\cdots \right)\left(1+\frac{1}{3}+\frac{1}{9}+\cdots \right)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{6}+\cdots$$
both of the first series are geometric and can be summed
$$1+\frac{1}{2}+\frac{1}{4}+\cdots =2$$
$$1+\frac{1}{3}+\frac{1}{9}+\cdots =\frac{3}{2}$$ so
$$1+\frac{1}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/827361",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Prove the sum $\sum_{n=1}^\infty \frac{\arctan{n}}{n}$ diverges. I must prove, that sum diverges, but...
$$\sum_{n=1}^\infty \frac{\arctan{n}}{n}$$
$$\lim_{n \to \infty} \frac{\arctan{n}}{n} = \frac{\pi/2}{\infty} = 0$$
$$\lim_{n \to \infty} \frac{ \sqrt[n]{\arctan{n}} }{ \sqrt[n]{n} } = \frac{1}{1} = 1$$
Cauchy's co... | Hint: $\displaystyle\lim_{N\to\infty}\sqrt[^N]{\dfrac\pi2}=1\neq0$. The same holds true if you replace $\dfrac\pi2$ with any other strictly positive
finite quantity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/827854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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$f(n)$ and $f(2^n)$ are co prime for all natural numbers $n$. Find all such polynomials. Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^n)$ are co prime for all natural numbers $n$.
| Here is a proof of non-existence of non-constant polynomials of this type: Suppose $f(x) = a_0 + a_1x + a_2x^2 + … + a_mx^m$ is such a polynomial (where $m>0$). We reach a contradiction.
If $a_0$ is even then $f(2)$ and $f(2^2)$ share a common factor of $2$. So $a_0$ must be odd. Thus, for every positive integer $k$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/828155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers?
Prove or disprove that if $t$ is a positive integer, $$f(x,y)=\dfrac{x^2+y^2}{xy-t},$$ then $f(x,y)$ has only finitely many distinct positive integer values with $x,y$ positive integer... | October 7, 2015. This is with $$ \frac{x^2 + y^2}{xy - t} > 0, $$
which I believe to be the intent of the question.
I proved finiteness, with an explicit bound that is not that bad.
This works. Note that the original question requires $xy> t.$ Otherwise we could have listed $x=1,y=1,t=2$ to get
$(x^2 + y^2)/ (xy-t) = ... | {
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"url": "https://math.stackexchange.com/questions/829228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Prove by induction that $(n+1)^2 + (n+2)^2 + ... + (2n)^2 = \frac{n(2n+1)(7n+1)}{6}$ Prove by induction that $$(n+1)^2 + (n+2)^2 + ... + (2n)^2 = \frac{n(2n+1)(7n+1)}{6}.$$
I got up to:
$n=1$ is true, and assuming $n=k$ prove for $n=k+1$.
Prove...
$$\frac{k(2k+1)(7k+1)+6(2(k+1))^2}{6} = \frac{(k+1)(2k+3)(7k+8)}{6}$$
I... | Alternative route (also with induction)
You can start proving inductively that: $$\sum_{k=1}^{n}k^{2}=\frac{1}{6}n\left(n+1\right)\left(2n+1\right)$$
Then have a look at: $$\sum_{k=n+1}^{2n}k^{2}=\sum_{k=1}^{2n}k^{2}-\sum_{k=1}^{n}k^{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/831521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Partial fraction (doubt) I have this partial fraction
$$\displaystyle\frac{1}{(2+x)^2(4+x)^2}$$
I tried to resolve using this method:
$$\displaystyle\frac{A}{2+x}+\displaystyle\frac{B}{(2+x)^2}+\displaystyle\frac{C}{4+x}+\displaystyle\frac{D}{(4+x)^2}$$
$$1=A(2+x)(4+x)^2+B(4+x)^2+C(4+x)(2+x)^2+D(2+x)^2$$
When x=-2
$$1=... | Assign different values to $x$, like $-1, 0, +1, +2$ and solve the linear system of 4 equations in 4 unknowns.
$$\left[\begin{array}{cccc}
1/1& 1/1& 1/3& 1/9\\
1/2& 1/4& 1/4& 1/16\\
1/3& 1/9& 1/5& 1/25\\
1/4& 1/16& 1/6& 1/36
\end{array}\right]
\left[\begin{array}{c}
A\\
B\\
C\\
D\end{array}\right]=
\left[\begin{array}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/833170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Factor the Expression completely$ (a+b)^2 - (a-b)^2$ I don't understand this question. The answer in the book is $4ab$, but how is that term a factor?
I was thinking along the line that this was a difference of squares example. $a^2-b^2 = (a+b)(a-b)$
My answer is $[(a+b)-(a-b)][(a+b)+(a-b)]$
What do I not understand? ... | $$(a + b)^2 - (a-b)^2 = (a+b)(a+b) - (a+b)(a-b) $$ $$= (a+b)((a+b) - (a - b)) $$ $$ = (a + b)(a + b - a + b) $$ $$= (a+b)(2b)$$
POST EDIT: $$(a+b)^2 - (a-b)^2 = a^2 +2ab + b^2 -(a^2 - 2ab + b^2) = 2ab - (-2ab) = 4ab$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/834111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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Shears using Matrix Methods Determine the equation of the image of the graph:
$$y=(x-1)^3 -2$$
after a shear of factor $1$ away from the $y$-axis, relative to the line $y=1$.
| You can parameterize the graph $y=(x-1)^3-2$ with the variable $t$ like so:
$$
\left(\begin{array}{c}
t\\
(t-1)^3-2
\end{array}\right)
$$
You can then transform this vector by translating it downwards one unit, performing the shear, then translating it back up. Because this involves doing a translation, we add an extra... | {
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"url": "https://math.stackexchange.com/questions/835824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$ This is a pretty straightforward question. I want to show $(x+y)^a > x^a + y^a$ for $x,y>0$ and $a>1$.
One way would be this. WLOG, suppose $x \leq y$. Then:
$(1+\frac{x}{y})^a >1+\frac{x}{y} \geq 1+(\frac{x}{y})^a$.
Noting that $\frac{1}{y^a}t_1>\frac{1}{y^a}t_2 \implie... | Write
$$ f(a) = \left( \frac{x}{x+y} \right)^a + \left( \frac{y}{x+y} \right)^a $$
It is clear that
$$ f(1) = \frac{x}{x+y} + \frac{y}{x+y} = 1 $$
It is also clear that
$$ f'(a) = \ln\left( \frac{x}{x+y} \right) \left( \frac{x}{x+y} \right)^{a-1} + \ln\left( \frac{y}{x+y} \right) \left( \frac{y}{x+y} \right)^{a-1} $$
w... | {
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"url": "https://math.stackexchange.com/questions/836337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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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.