Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Prove that ${e\over {\pi}}\lt{\sqrt3\over{2}}$ without using a calculator. I have been working on a known question for a long time (this is "Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator") during this time I realized the ${e\over {\pi}}\lt{\sqrt3\over{2}}$.
I have no solution so far. Do you have any id... | Calculation without any computer, only with patience. :-D
$\displaystyle e<\frac{\sqrt{3}}{2}\pi\enspace$ is equivalent to $\enspace\displaystyle \frac{2}{9}\sum\limits_{n=0}^\infty \frac{2^n}{n!} =\frac{2e^2}{9}<\frac{\pi^2}{6}=\zeta(2)$
Case $\,(A)\,$ :
We have $\enspace\displaystyle \prod\limits_{k=0}^n \frac{k+9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
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Compute $\lim_{x\rightarrow -\infty}\left(\sqrt{x^2-6x+9}+x-1\right)$ I note that $\sqrt{x^2-6x+9}=|x-3|$. Splitting upp the limit into cases gives
*
*$x\geq 3:$
$$\lim_{x\rightarrow -\infty}\left(\sqrt{x^2-6x+9}+x-1\right)=\lim_{x\rightarrow -\infty}(|x-3|+x-1)=2\lim_{x\rightarrow -\infty}(x-2)=-\infty.$$
*
*$... | Here is a different approach, setting $x=-t$, the limit becomes:
$$\lim_{t \to \infty} \sqrt{t^2+6t+9} - t - 1$$
Since $t^2+6t+9=(t+3)^2$, we can say for positive t-values that $\sqrt{(t+3)^2}=t+3$, your limit expression becomes $t+3-t-1=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2440429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Prove the limit $\lim_{x\to 1+}\frac{1}{\sqrt{x}}=1$, using epsilon-delta definition. $$\lim_{x\to 1+}\frac{1}{\sqrt{x}}=1$$
The proof that I have:
Let $\varepsilon > 0$, we must show that
$$\exists \delta >0: 0<x-1<\delta \Rightarrow \left | \frac{1}{\sqrt{x}}-1\right|<\epsilon$$
So usually, when doing these $\varep... | Corrections:
We assume that $x$ is chose from $0<x-1<1$:
$$
\begin{align}
0<&x-1<1\\
1<&x<2\\
1<&\sqrt{x}<\sqrt{2}\\
\frac{1}{\sqrt{2}}<&\frac{1}{\sqrt{x}}<1\\
2<&\sqrt{x}+1<\sqrt{2}+1\\
\frac{1}{\sqrt{2}+1}<&\frac{1}{\sqrt{x}+1}<\frac{1}{2}
\end{align}
$$
$$\left | \frac{1}{\sqrt{x}}-1\right|= \left | \frac{x-1}{\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Quadratics and roots Consider the equation (E): $$x^2 - (m+1)x+m+4=0$$ where $m$ is a real parameter
determine $m$ so that $2$ is a root of (E) and calculate the other root.
This is the question.
What I did was basically this:
Let the sum of root 1 and root 2 be $S$ and their product $P$
Let $x_2 = a ; x_1=2$(given)... | If $2$ is a root of $x^2-(m+1)x+(m+4)=0$ then
\begin{align}
&2=\frac{(m+1)\pm\sqrt[\;2]{(m+1)^2-4\cdot (m+4)}}{2}
\\
\Leftrightarrow&
4=(m+1)\pm \sqrt[\;2]{m^2-2m-15}
\\
\Leftrightarrow&
-m+3 =\pm \sqrt[\;2]{m^2-2m-15}
\\
\Leftrightarrow&
(-m+3)^2 =\left(\pm \sqrt[\;2]{m^2-2m-15}\right)^2
\\
\Leftrightarrow&
(-m+3)^2 =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2441732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Probability of a target being hit I don't have the answer of the following question so I wanted to cross check my solution.
$A$ can hit a target 3 times in 5 shots, $B$ 2 times in 5 shots and $C$ 3 times in 4 shots. Find the probability of the target being hit at all when all of them try.
my method
P(target being hit... | Let $a$ indicate that person $A$ hit the target, $b$ indicate that $B$ hit the target, and $c$ that $C$ hit the target.
We expect $a$ to occur $3/5$ of the time, $b$ to occur $2/5$ of the time and $c$ to occur $3/4$ of the time.
The event “at least one hits the target” is the complement of “none of them hits the target... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2442777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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No. of isosceles triangles possible of integer sides with sides $\leq n$ Prove that the no. of isosceles triangles with integer sides, no sides exceeding $n$ is $\frac{1}{4}(3n^2+1)$ or $\frac{3}{4}(n^2)$ according as n is odd or even, n is any integer.
How to do it? I found that under these conditions no. of triangles... | ${n \choose 2}$ is the number of triplets $(k,k, m)$. But not all triplets can be triangles. To be a triangle i) $m < k + k = 2k$ and ii) $k < k + m$. (i) is a essential, ii) is trivially redundant).
So we need to find all possible triplets $(k,k,m)$ where $k,m \le n$ and $m < 2k$. As $m$ is an integer, that means ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2443341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Solve the equation $2x^2-[x]-1=0$ where $[x]$ is biggest integer not greater than $x$. I' ve tried with $x^2 = {[x]+1\over 2}$ so $x$ is a square root of half integer. And know? What to do with that?
| we know $0\leq x-\lfloor x \rfloor <1 $
$$\quad{2x^2-\lfloor x \rfloor-1=0 \to \lfloor x \rfloor=2x^2-1
\\so\\0\leq x-(2x^2-1) <1\to \\
\begin{cases}0\leq x-(2x^2-1) \to & -(x-1)(2x+1)\geq 0 & (*)\\ x-(2x^2-1) <1 \to & x(1-2x)<0 & (**)\end{cases}
\\\begin{cases} (*) \to & x\in[-\frac12,1]\\ (**)\to &x\in (-\infty,0)\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2446000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Proof verification: $ \overline{v}_{1} = \overline{v}_{2} \iff v_{1} - v_{2} \in W \iff v_{2} - v_{1} \in W.$
Let W be a subspace of V. Then for $\overline{\textbf{v}}_{i} =
\overline{\textbf{v}}_{i} + W $,
$$ \overline{\textbf{v}}_{1} =
\overline{\textbf{v}}_{2} \iff \textbf{v}_{1} - \textbf{v}_{2} \in W
\iff \... | Yes, it is correct. Note that there is a typo at the statement. You wrote “$\overline{\mathbf{v}}_i=\overline{\mathbf{v}}_i+W$”, but I guess that you meant “$\overline{\mathbf{v}}_i=\mathbf{v}_i+W$”.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2446124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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$\frac{a^3}{b^2} + \frac{b^3}{c^2} + \frac{c^3}{a^2} \geq 3 \, \frac{a^2 + b^2 + c^2}{a + b + c}$ Proposition
For any positive numbers $a$, $b$, and $c$,
\begin{equation*}
\frac{a^3}{b^2} + \frac{b^3}{c^2} + \frac{c^3}{a^2}
\geq 3 \, \frac{a^2 + b^2 + c^2}{a + b + c} .
\end{equation*}
I am requesting an elementary, alg... | we have $$\frac{a^4}{ab^2}+\frac{b^4}{bc^2}+\frac{c^4}{ca^2}\geq \frac{a^2+b^2+c^2)^2}{ab^2+bc^2+ca^2}\geq \frac{3(a^2+b^2+c^2)}{a+b+c)}$$
the last is true, since$$a(a-c)^2+b(a-b)^2+c(b-c)^2\geq 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2446312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Prove that $\frac{ab}{a+b} + \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$ $$\frac{ab}{a+b} + \frac{cd}{c+d} \leq \frac{(a+c)(b+d)}{a+b+c+d}$$
I tried applying a.m. g.m inequality to l.h.s and tried to find upper bound for l.h.s and lower bound for r.h.s but i am not getting answer .
| By C-S we obtain:
$$\frac{ab}{a+b}+\frac{cd}{c+d}=a+c+\left(\frac{ab}{a+b}-a\right)+\left(\frac{cd}{c+d}-c\right)=$$
$$=a+c-\left(\frac{a^2}{a+b}+\frac{c^2}{c+d}\right)\leq a+c-\frac{(a+c)^2}{a+b+c+d}=\frac{(a+c)(b+d)}{a+b+c+d}.$$
Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2448601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Basic arithmetic with matrices We have matrices.
$$A=\begin{bmatrix}
-2 & 0 \\
-5 & 6 \\
\end{bmatrix}
B^{-1}=\begin{bmatrix}
-7 & 8 \\
2 & -8 \\
\end{bmatrix}
C=\begin{bmatrix}
-15 & -2 \\
-8 & -14 \\
\end{bmatrix}
$$
We need to solve matrix $X$ from equation:
$$A^{-1}XB-C=0$$
$$X=AB^{-1}+C$$
$$X=\begin{bmatrix}
-2 &... | The equation
$X=AB^{-1}+C \tag 1$
does not follow from
$A^{-1}XB-C= 0; \tag 2$
instead, we have
$A^{-1}XB = C, \tag 3$
$XB = AC,\tag 4$
$X = ACB^{-1}; \tag 5$
if we now perform the indicated matrix arithmetic we arrive at
$X = \begin{bmatrix} -2 & 0 \\ -5 & 6 \end{bmatrix} \begin{bmatrix} -15 & -2\\-8 & -14 \end{bmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
a definite integral identity? is it true? $\int_{a}^{b} f(x) dx $ i came across this identity , i don't know if it is true or not
$$\int_{a}^{b} f(x) \ \mathrm{d}x = (b-a) \sum_{n=1}^{\infty} \sum_{k=1}^{2^n - 1} \dfrac{(-1)^{k+1}}{2^{n}} f \left( a+ \left(\frac{b-a}{2^n}\right) k \right)$$
i tried to use the Riemann S... | Note that:
$$
\int_a^bf(x)dx=\lim_{N \to \infty}S_N \\
S_N=\sum_{i=1}^{2^N-1} \frac{b-a}{2^N}f(a+\frac{b-a}{2^N}i) =
\sum_{n=1}^N \sum_{k=1}^{2^n-1}\frac{b-a}{2^n}(-1)^{k+1}f(a+\frac{b-a}{2^n}k)
$$
Prove the last identity by induction:
For $N$=1, trivial.
For $N\to N+1$:
$$
\sum_{n=1}^{N+1} \sum_{k=1}^{2^n-1}\frac{b-a}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Solving the equation $c=\dfrac{x^2+y^2-1}{x^2+(y+1)^2}$ Solving the equation $c=\dfrac{x^2+y^2-1}{x^2+(y+1)^2}$
$$c=\dfrac{x^2+y^2-1}{x^2+(y+1)^2}$$
$$c{x^2+c(y+1)^2}={x^2+y^2-1}$$
$$c{x^2+cy^2+2cy+c}={x^2+y^2-1}\text{ [expanded]}$$
$$1+c=x^2-cx^2+y^2-cy^2-2cy\text{ [moved to other side]}$$
$$1+c=(1-c)x^2+\color{red}{(... | Here
$1+c=(1-c)x^2+\color{red}{(1-c)(y^2-\dfrac{2cy}{c-1}+(\dfrac{2c}{c-1})^2)-(\dfrac{2c}{c-1})^2}\text{ [completed square]}$
It's not equal to the precedent line..... sign problem and you didnt add the correct square....
By the way put $c=2$ in the first equation, you get :
$x^2+(y+2)^2=1$ which is a circle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453143",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Prove using some form of induction that $T(n) = O(n^2)$ given recurrence relation
Here is a recursively defined function where $c \ge 0$.
$T(n) = c$ if $n = 0$
$T(n) = c$ if $n = 1$
$T(n) = 2T(n-1) - T(n-2) + 2$ if $ \ n \geq 2$
Prove using some form of induction that $ \ T(n) = O(n^2)$
My attempt:
We have to show t... | Some corrections are needed. In the induction hypothesis you used the inequality $-T(n-2) \le -c(n-2)^2$ or $T(n-2) \ge c(n-2)^2$ which is not true.
Hint. For $n\geq 1$, let $S(n):=T(n) -T(n-1)$. Then $S(1)=0$ and for $n\geq 2$
$$S(n)=T(n) -T(n-1)=T(n-1) -T(n-2)+2=S(n-1) + 2.$$
Hence (by induction?), for $n\geq 1$, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2459930",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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The sum of series $\frac14+\frac{1\cdot3}{4\cdot6}+\cdots$ The problem I have here today is the following;
$$\frac{1}{4}+\frac{1\cdot3}{4\cdot6}+\frac{1\cdot3\cdot5}{4\cdot6\cdot8}+\cdots$$ the problem is exactly phrased like this (I can't say that the $\infty$ sign is a bit unnecessary at the end),
My Attempts
We can ... | Amusingly, since the sum is $1$, you can actually write this as a question about a random number.
For each $n$, we flip an unfair coin $C_n$ with heads having probability of $\frac{1}{2(n+1)}$.
Let $X$ be the random variable which is $n$ if $C_n$ came up heads and for each $i<n$, $C_i$ came up tails.
Since $\prod_{k=1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
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Anti-derivative of a function that involves poly-logarithms. Let $n\ge 1$ be an integer and let $0 < z < a$ be real numbers.
Let $Li_n(x):= \sum\limits_{l=1}^\infty z^l/l^n$ by the polylogarithm of order $n$.
The question is to find the following anti-derivative:
\begin{equation}
{\mathfrak J}^{(n)}_a(z):=\int\frac{L... | Here we provide an answer for $n=3$. The idea is to expand the function to be sought for in a series about the value $a=1$. Clearly we have:
\begin{equation}
-{\mathfrak J}^{(n)}_a(a-x)=\sum\limits_{m=0}^\infty \int \frac{Li_3(x)}{(1-x)^{m+1}}dx \cdot (1-a)^m
\end{equation}
Now, using integration by parts we can derive... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Equation of the tangent to a graph where the point is not on the graph. If $f(x)=(x+1)^{3/2}$, provided $x\geq -1$, I am asked to find the equation of all tangent lines to $f(x)$ at the point $(\frac{4}{3},3)$.
Simple enough. I first took the derivative which is:
$$f'(x)= \frac {3\sqrt{x+1}}{2}$$
Since $(\frac{4}{3},3)... | $$3-\frac{1}{2} k\sqrt{k+1} - 3 \sqrt{k+1} =0$$
$$3=\frac{1}{2}k\sqrt{k+1} + 3 \sqrt{k+1}$$
$$3=\sqrt{k+1}(\frac{1}{2}k+3)$$
$$9=(k+1)(\frac{1}{4}k^{2}+9+3k)$$
$$\frac{1}{4}k^3 + \frac{13}{4}k^2+12k=0$$
$$k(\frac{1}{4}k^2 + \frac{13}{4}k+12)=0$$
from here you now one answer is $k=0$ and others are those who make
$$\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if $f(x)= e^{-1/x^2}\sin{\frac{1}{x}}$ for $x\neq0$ and $f(0)=0$, then $f^{(k)}(0)=0$ for all $k$. This question is from Spivak's Calculus (3rd ed) 18-41:
Prove that if $f(x)= e^{-1/x^2}\sin{\frac{1}{x}}$ for $x\neq0$ and $f(0)=0$, then $f^{(k)}(0)=0$ for all $k$.
Solution is:
I need someone to explain ho... | I do not know if this is correct, but here is one approach to prove it.
Given that
$$f^{\left( k \right)}\left( x \right) = e^{-\frac{1}{x^2}}\left[ \sum\limits_{i = 1}^{3k} \dfrac{a_i}{x^i} \sin \dfrac{1}{x} + \sum\limits_{i = 1}^{3k} \dfrac{b_i}{x^i} \cos \dfrac{1}{x} \right]$$
We have from here,
$$f^{\left( k \righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2472155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Non zero solutions of second order equation by solving for $a$ Im stuck on this problem:
Let a be a real constant. Consider the equation
$y''+5y'+ay=0$ with boundary conditions $y(0)=0$ and $y(3)=0$
For certain discrete values of $a$, this equation can have non-zero solutions. Find the three smallest values of $a$ for ... | The characteristic equation of the homogeneous ODE $y''+5y'+ay=0$ is $r^2+5r+a=0$. The general solution of the equation is $y(x) = \lambda_+e^{r_+ x} + \lambda_-e^{r_- x}$, where
$$
r_\pm = -\frac{5}{2}\pm\sqrt{\frac{25}{4}-a}
$$
and the constants $\lambda_+$, $\lambda_-$ are deduced from the boundary values. Of course... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Pauli matrices and the complex number matrix representation The three spin Pauli matrices are:
$
\sigma_1 = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},
\sigma_2 = \begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix},
\sigma_3 = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
$
According to problem 2.2.3 (Mathematica... | The mapping $a +ib \Rightarrow \begin{pmatrix}
a & b \\
-b & a
\end{pmatrix}$ where $ a,b \in \mathbb{R}$ does not map any complex number to a Pauli matrix. The Pauli matrices are not of the right form. So saying that $1$ maps to $\sigma_2$ (or even worse, that it is equal to it) is meaningless.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2475806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Bag marbles with three colors and probability A bag contains one red, two blue, three green, and four yellow balls. A sample of three balls is taken without replacement. Let $Y$ be the number of yellow balls in the sample. Find the probability of $Y=0$, $Y=1$, $Y=2$ $Y=3$
Attempt 1
all three are yellow would be
$$ ... | We have four yellow balls and $1 + 2 + 3 = 6$ balls that are not yellow. The number of ways we can select exactly $k$ yellow balls and $3 - k$ balls that are not yellow is
$$\binom{4}{k}\binom{6}{3 - k}$$
Since there are $\binom{10}{3}$ possible selections of three of the ten balls, the desired probabilities are
\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2476098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to show that $\int_{0}^{\pi}\sin^3(x){\mathrm dx\over (1+\cos^2(x))^2}=1?$ How can we show that $(1)$
$$\int_{0}^{\pi}\sin^3(x){\mathrm dx\over (1+\cos^2(x))^2}=1?\tag1$$
$\sin^3(x)={3\over 4}\sin(x)-{1\over 4}\sin(3x)$
$1+\cos^2(x)=2-\sin^2(x)$
$$\int_{0}^{\pi}[{3\over 4}\sin(x)-{1\over 4}\sin(3x)]{\mathrm dx\ove... | Well, we have:
$$\mathscr{I}_{\space\text{n}}:=\int_0^\text{n}\frac{\sin^3\left(x\right)}{\left(1+\cos^2\left(x\right)\right)^2}\space\text{d}x\tag1$$
Substitute $\text{u}:=\cos\left(x\right)$:
$$\mathscr{I}_{\space\text{n}}:=\int_1^{\cos\left(\text{n}\right)}\frac{\text{u}^2-1}{\left(1+\text{u}^2\right)^2}\space\text{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2476520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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Inequality question: If $a + b + c =1$, what is the minimum value of $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$. If $a + b + c =1$, what is the minimum value of $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$. I've tried AM-HM but it gave $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 9$ which gives $\frac{1}{a^2} + \f... | $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$ = $\frac{c}{abc} + \frac{a}{abc} + \frac{b}{abc}$ = ${\frac{1}{abc}}$
We know that AM${\ge}$GM
${\frac{a+b+c}{3}}$ ${\ge} \sqrt[3]{ abc}$
${\frac{1}{3}}$ ${\ge} \sqrt[3]{ abc}$
${\frac{1}{27}}$ ${\ge} { abc}$
${\frac{1}{abc}}$ ${\ge} 27$
so minimum value is 27
This minimum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove that series $(10n+4)^2+1$ contains infinitly many composite numbers Given sequence is
$$4^2+1,\ 14^2+1,\ 24^2+1,\ 34^2+1...$$
How to show that there are infinite amount of composite numbers?
| If you take any prime $p$ such that $p\mid (10n+4)^2+1$ then $p\mid (10(n+kp)+4)^2+1$ because $$(10(n+kp)+4)^2+1=(10n+4+kp)^2+1=(10n+4)^2+2kp(10n+4)+k^2p^2+1=(10n+4)^2+1+p(2k(10n+4)+k^2p)$$
And trivially $$p\mid p(2k(10n+4)+k^2p)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2478560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Showing that $\frac{\sin(a_{n-1}) + 1}{2}$ is a Cauchy sequence In my homework set, I have the following question:
Show that $$ a_n = \frac{\sin(a_{n-1}) + 1}{2}, \quad a_1=0 $$
satisfies the definition of Cauchy sequence.
As we went over the concept of Cauchy sequences a bit too quickly in class, I'm puzzled abou... | Hint. Note that the function $f:\mathbb{R}\to \mathbb{R}$,
$$f(x)=\frac{\sin(x)+1}{2}$$
is a contraction: for any $x,y\in \mathbb{R}$ there exists $t$ between $x$ and $y$ such that
$$|f(x)-f(y)|\leq \left|\frac{\cos(t)}{2}\right||x-y|\leq \frac{|x-y|}{2}\tag{1}$$
where we used the Mean value theorem. Hence, for $n\geq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2481450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Parametric equation of the intersection between $x^2+y^2+z^2=6$ and $x+y+z=0$ I'm trying to find the parametric equation for the curve of intersection between $x^2+y^2+z^2=6$ and $x+y+z=0$. By substitution of $z=-x-y$, I see that $x^2+y^2+z^2=6$ becomes $\frac{(x+y)^2}{3}=1$, but where should I go from here?
| The equation
$$
x^2 + y^2 + z^2 = 6
$$
gives a sphere around the origin with radius $R=\sqrt{6}$.
$$
0 = x + y + z = (1,1,1) \cdot (x,y,z)
$$
gives a plane $H$ with normal vector $(1,1,1)^\top$ including the origin.
The intersection is a circle of radius $R$ on that plane $H$, with the origin as midpoint.
GeoGebra seem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Taylor expansion of $\ln(1 + \frac{2^x}{n})$ I have a function $f = \ln(1 + \frac{2^x}{n})$, where $n \to \infty$ and $x \in (0, 1)$.
I want to apply Taylor expansion at $a = 0$ to $f$.
I get $f = \dfrac{2^x}n-\dfrac{4^x}{2n^2}+o(\dfrac{4^x}{2n^2})$
My question is for what $x$ this approximation is true and why?
Wolfr... | Let $f(x) = \ln\left( 1 + \frac{2^{x}}{n}\right)$ then
\begin{align}
f'(x) &= \frac{\ln2 \, 2^{x}}{n + 2^{x}} \\
f''(x) &= \frac{n \, \ln^{2}2 \, 2^{x}}{(n+2^{x})^2} \\
f'''(x) &= \frac{n(n-1) \, \ln^{3}2 \, 2^{x} \, (n-2^{x})}{(n+2^{x})^3}
\end{align}
which leads to
$$f_{n}(x) = \ln\left(1 + \frac{1}{n}\right) + \left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Prove: if |$z|=1$ then $|iz-\overline{z}|\leq 2$
Prove: if |$z|=1$ then $|iz-\overline{z}|\leq 2$
$$|iz-\overline{z}|=|i(a+bi)-(a-bi)|=|-(a+b)+(a+b)i|=\sqrt{(a+b)^2+(a+b)^2}=\sqrt{2(a+b)^2}$$
On the other hand $|z|=1\iff \sqrt{a^2+b^2}=1\iff a^2+b^2=1$
So it seems that $\sqrt{2(a+b)^2}\leq \sqrt{2}$ maybe it is $|iz-... | One way to proceed from your point is by using the inequality $2ab\le a^2+b^2$ $$0\le (a+b)^2=1+2ab\le1+a^2+b^2= 2$$ And thus your quantity ends up being in the interval $\left[\sqrt0,\sqrt4\,\right]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2489684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
What is $\lim_\limits{x\to 0}\frac{x}{1-\cos{x}}$ equal to? At 29:30 in lecture 8 of UMKC's Calculus I course, the instructor makes the claim that its limit is equal to zero mentioning that he proved this result earlier in the lecture. The thing is that I watched the entire lecture and yet never actually saw him do tha... | Write $\dfrac{x}{1-\cos x} = \dfrac{x(1+\cos x)}{1-\cos^2 x}= \dfrac{x^2(1+\cos x)}{x\sin^2 x}= \left(\dfrac{x}{\sin x}\right)^2\cdot \dfrac{1+\cos x}{x}$. The limit does not exist because $\dfrac{1+\cos x}{x} \to +\infty$ and $-\infty$ when $x \to 0^{+}, 0^{-}$ respectively.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2491894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Prove that if $p|x^p+y^p$ then $p^2|x^p+y^p$ I can show that $5|x^5+y^5$, by considering $(x+y)^5$ and using binomial expansion. But I am not sure how to show that $25|x^5+y^5$.
More generally, if p is a prime and $p>2$, how do I prove that if $p|x^p+y^p$ then $p^2|x^p+y^p$?
| Notice for all prime $p$ and integer $n$, $p | n^p - n$.
In particular, $p | x^p - x$ and $p | y^p - y$. This means
$$p|x^p + y^p\quad\implies\quad p|x+y$$
Write $x+y$ as $mp$ for some integer $m$, we have
$$\begin{align}
x^p + y^p
&= x^p + (mp-x)^p\\
&= x^p + (-x)^p + \binom{p}{1}(mp)(-x)^{p-1} + (mp)^2\left(\sum_{k=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2494388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Envelope of a family of curves
Show that the envelope of the family of curves $$\frac{x}{a}+\frac{y}{4-a}=1$$ is the parabola $$\sqrt{y}+\sqrt{x}=2$$
I know how we can get the envelope, but I could not get the required relation. I differentiated the family of curves w.r.t $a$. I got
$$a=\frac{1}{2}(x-y+4)$$
I then s... | Maybe this is a simpler way to approach the problem.
I will assume that ${0\leq a \leq 4}$.
Define $D$ as the union of the family of curves $\begin{align*}\frac{x}{a}+\frac{y}{4-a}=1\end{align*}$. Then there exists a function $~{f(x)}$ defined on the interval $~x\in [0,~4]$ such that the region $~D$ can be written as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Jordan canonical form and reordering
Consider the Jordan canonical form below \begin{equation*} J =
\begin{pmatrix} J_2(\lambda_1)&0& 0& 0\\ 0&J_1(\lambda_2) &0& 0\\ 0&
0& J_3(\lambda_1)& 0\\ 0 &0 &0 &J_2(\lambda_2) \end{pmatrix}
\end{equation*} \begin{equation*}
= \begin{pmatrix} \lambda_1& 1& 0& 0& 0& 0& 0& 0\\ 0... | Let $A \in \mathbb{C}^{n \times n}$ have the following two Jordan Decompositions:
\begin{align*}
A = X_1 J_1 X_1^{-1}, \quad A = X_2 J_2 X_2^{-1}.
\end{align*}
Here, $J_1, J_2 \in \mathbb{C}^{n\times n}$ is block-diagonal, and $X \in \mathbb{C}^{n\times n}$ is invertible. Then we have
\begin{align*}
X_1 J_1 X_1^{-1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Integral argument to a polynomial identity One can show that
$$\frac{(X^{15} - 1)(X-1)}{(X^5-1)(X^3-1)} = P^2 + 15 Q^2$$ with $P,Q\in\mathbf{Z}\left[\frac{1}{2}\right][X]$, by simply grouping terms etc. Is it possible to show that such an identity exists by arguments involving $\mathbf{Z}\left[\sqrt{15}\right]$ or othe... | Nice question! The LHS clearly is the cyclotomic polynomial $\Phi_{15}(x)$, and by considering its factorization over the ring of integers of $\mathbb{Z}[\sqrt{5}]$ we have
$$ \Phi_{15}(x) = \frac{1}{4}\left(2+\left(-1+\sqrt{5}\right) x+\left(1-\sqrt{5}\right) x^2+\left(-1+\sqrt{5}\right) x^3+2 x^4\right)\cdot\left(2-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Binomial summation problem
Show that
$$ \frac{\binom{n}{0}}{1} - \frac{\binom{n}{1}}{4} +\dots + (-1)^n \frac{\binom{n}{n}}{3n+1} = \frac{3^n \cdot n!}{ 1\cdot 4\cdot 7\cdots(3n+1)}.$$
I don't know how to proceed in such type of problems. Any help or hint will be much appreciated.
| Let
$$a_n:=\frac{\binom{n}{0}}{1} - \frac{\binom{n}{1}}{4} +... + (-1)^n \frac{\binom{n}{n}}{3n+1} =\sum_{k=0}^n\frac{(-1)^k\binom{n}{k}}{3k+1}$$
Then for $n\geq 1$,
\begin{align}
a_n&=1+\sum_{k=1}^n\frac{(-1)^k\frac{n}{k}\binom{n-1}{k-1}}{3k+1}
=1+n\sum_{k=1}^n(-1)^k\binom{n-1}{k-1}\left(\frac{1}{k}-\frac{3}{3k+1}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2498077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Limit with n roots I have been trying to practice computing limits and this one came across:
$$\lim_{x\to a} \frac{\sqrt[n]{x}-\sqrt[n]{a}}{x-a}$$
I tried L'Hopital and I got this:
$$\lim_{x\to a}{\frac{x^{\frac{1-2n}{n}}\left(1-n\right)}{n}}$$
But I should get as the solution of the limit $\frac{\sqrt[n]{a}}{an}$
Any... | $$\lim _{ x\to a } \frac { \sqrt [ n ]{ x } -\sqrt [ n ]{ a } }{ x-a } =\lim _{ x\to a } \frac { \left( \sqrt [ n ]{ x } -\sqrt [ n ]{ a } \right) \left( \sqrt [ n ]{ { x }^{ n-1 } } +\sqrt [ n ]{ { x }^{ n-2 }a } +\sqrt [ n ]{ { x }^{ n-3 }{ a }^{ 2 } } +...+\sqrt [ n ]{ { a }^{ n-1 } } \right) }{ \left( x-a \righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2500296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
Local extremes of: $f(x,y) = (x^2 + 3y^2)e^{-x^2-y^2}$ I am looking to find the local extremes of the following function:
$$f(x,y) = (x^2 + 3y^2)e^{-x^2-y^2}$$
What have I tried so far?
*
*Calculate the partial derivatives:
$$\frac{\partial f}{\partial x} = 2x(e^{-x^2-y^2}) + (x^2+3y^2)(e^{-x^2-y^2})(-2x)$$
$$=-2(... | In 1D (single variable Calculus), a stationary-critical point is a point $c$ such that $f'(c)=0$. At $c$ the gradient of $f$ is zero (there is no change).
In 2D, a stationary-critical point is a point $(a, b)$ such that the gradient in $x$ direction and $y$ direction are both zero (at that same point $(a,b)$, simultan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$a_1 = a_2 = 1$ and $a_n = \frac{1}{2} \cdot (a_{n-1} + \frac{2}{a_{n-2}})$. Prove that $1 \le a_n \le 2: \forall n \in \mathbb{N} $
Let $a_n$ be a sequence satisfying $a_1 = a_2 = 1$ and $a_n = \frac{1}{2} \cdot (a_{n-1} + \frac{2}{a_{n-2}})$. Prove that $1 \le a_n \le 2: \forall n \in \mathbb{N} $
Attempt at soluti... | Assume $x,y\in [1,2].$ Then $x\le 2$ and $y\ge 1\implies \dfrac{1}{y}\le 1\implies \dfrac{2}{y}\le 2.$ Thus
$$\dfrac12\left(x+\dfrac2y\right)\le \dfrac12(2+2)=2.$$ On the other hand $x\ge 1$ and $1\le y\le 2\implies \dfrac{1}{y}\ge\dfrac12 \implies \dfrac{2}{y}\ge 1.$ Thus
$$\dfrac12\left(x+\dfrac2y\right)\ge \dfrac1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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Finding $S_{n}$ for this geometric series? Here is what was given to me:
$t_{1}$ = $\frac {1}{256}$
$r$ = $-4$
$n$ = $10$
The formula I used is $S_{n}$ = $\frac {t_{1} (r^{n} - 1)}{r - 1}$
Here's what I did:
$S_{10}$ = $\frac {\frac{1}{256} (-4^{10} - 1)}{-4 - 1}$
$S_{10}$ = $\frac {\frac{1}{256} (-1048576 - 1)}{-5}$
$... | Given:
$t_{1}$ = $\frac {1}{256}$
$r$ = $-4$
$n$ = $10$
$S_{n} = \frac {t_{1} (r^{n} - 1)}{r - 1}$
then $(-4)^{10} = (-1)^{10} \cdot 1048576 = 1048576$ and
\begin{align}
S_{10} &= \frac{1}{256} \, \frac{(-4)^{10} - 1}{-4 -1} \\
&= \frac{1}{256} \, \frac{1048575}{-5} = - \frac{209715}{256}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2508543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Proving $(1+a)^n\ge 1+na+\frac{n(n-1)}{2}a^{2}+\frac{n(n-1)(n-2)}{6}a^3$ for all $n\in\mathbb N$ and all $a\ge -1.$ I was asked to prove the the following without induction. Could someone please verify whether my proof is right? Thank you in advance.
For any real number $a\ge -1$ and every natural number n, the statem... | Also, we can use the following reasoning.
For $n\in\{1,2,3\}$ it's an identity.
Let $n\geq4$ and $$f(a)= (1+a)^n-1-na-\frac{n(n-1)}{2}a^{2}-\frac{n(n-1)(n-2)}{6}a^3.$$
Thus,
$$f'(a)=n(1+a)^n-n-n(n-1)(n-2)a-\frac{n(n-1)(n-2)}{2}a^2,$$,
$$f''(a)=n(n-1)(1+a)^{n-2}-n(n-1)-n(n-1)(n-2)a$$ and
$$f'''(a)=n(n-1)(n-2)(1+a)^{n-3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2508809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prove that $a_n=(1-\frac{1}{n})^n$ is monotonically increasing sequence I try to solve it Bernoulli inequality but it too complicated, am I missing something easier?
My try-
$$\frac{a_n}{a_{n+1}}=\frac{(1-\frac{1}{n})^n}{(1-\frac{1}{n+1})^{n+1}}\\=(\frac{1}{1-\frac{1}{n+1}})(\frac{\frac{n-1}{n}}{\frac{n}{n+1}})^n\\=(\f... | $a_n:= (1-1/n)$, $n\in \mathbb{Z+}.$
$a_{n+1} = (1-1/(n+1)).$
$a_n \lt a_{n+1}.$
$\rightarrow:$
$(a_n)^n < (a_{n+1})^n.$
Since $1> a_n >0 :$
$a_n (a_n)^n < (a_{n+1})^n, $
$(a_n)^{n+1} < (a_{n+1})^n.$
Set $s:= \dfrac{1}{n(n+1)} >0$,
Note : $ 0 <s<1$.
Consider $f(x) := x^s,$ $x >0$, real.
$f'(x) = s\dfrac{1}{x^{1-s}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
prove the angle is 90 degrees Given the trianle ABC,draw AD, where D is the middle of BC.If the angle BAD is 3 times the angle DAC and the angle BDA is 45 degrees,then prove that the angle BAC is 90 degrees.
I tried to draw a parallel line to BA and compare congruent trianges ,after extending AD to meet the parallel li... | Here is a trigonometric proof.
Let $x=\angle DAC$; then $3x=\angle BAD$. Let $y=BD=DC$. Then some angle chasing gives $\angle ABC=135^{\circ}-3x$ and $\angle ACB=45^{\circ}-x$. By the Law of Sines,
\begin{align*}
\frac{y}{\sin 3x} &= \frac{AD}{\sin(135^{\circ}-3x)} \\
\frac{y}{\sin x} &= \frac{AD}{\sin(45^{\circ}-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2520796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solution of a Volterra integral equation Given the Volterra integral equation
$$y(x)=x-\int_0^x xt^2 y(t)\,dt$$
How do I solve it? Predicting $y$ using iteration is seeming difficult. Please help.
| We start from this equation:
$$y(x) = x\left(1 - \int_0^x t^2 y(t) dt \right)$$
Differentiating both sides, we get:
$$\frac{dy(x)}{dx} = \left(1 - \int_0^x t^2 y(t) dt \right) - x^3y(x).$$
Notice that:
$$y(x) = x\left(1 - \int_0^x t^2 y(t) dt \right) \Rightarrow \left(1 - \int_0^x t^2 y(t) dt \right) = \frac{y(x)}{x}.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2521590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the : $\sum_{i=1}^{n}\dfrac{1}{i!}=\text{?}$
Find the :
$$\sum_{i=1}^n \frac{1}{i!}=\text{?}$$
For ex :
$$\sum_{i=1}^{100}\dfrac{1}{i!}=\text{?}$$
My Try :
$$\frac{1}{2\times 1}+\frac{1}{1 \times 2 \times 3 }=\frac{3(1)+1}{1 \times 2 \times 3 }$$
$$\frac{4}{1 \times 2 \times 3 }+\frac{1}{1 \times 2 \times 3 \... | The "closed form" is $\frac{e \Gamma(n+1,1)}{n!} - 1$ where $\Gamma(\cdot,\cdot)$ is the incomplete Gamma function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2523650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How many ways to form 3 unordered partitions from n-element set? Let's say a set $S$ has $n$ elements, and it needs to be partitioned into $3$ different, unordered partitions. How do I obtain a general formula for this?
I think I can calculate it if I know the value of $n$.
For example, if $n=3$ then it's partitioned ... | The number of partitions of a set of size $n$ into $k$ non-empty sets is counted by the Stirling numbers of the second kind. As explained in the link, they satisfy the recurrence
$$ \left\{ \begin{array}{c} n \\ k \end{array} \right\} = k\left\{ \begin{array}{c} n - 1 \\ k \end{array} \right\}+\left\{ \begin{array}{c} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2525013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Coefficient of $x^{n-2}$ in $(x-1)(x-2)(x-3)\dotsm(x-n)$ Question
Find the coefficient of $x^{n-2}$ in the expression $$(x-1)(x-2)(x-3)\dots(x-n)~~.$$
My approach
The coefficient of $x^n$ is $1$. The coefficient of $x^{n-1}$ is $- \frac{n(n+1)}{2}$
But I cannot proceed from here.
I would appreciate any help.
| Finding the coefficient of $x^{n-2}$ requires picking $2$ terms from the product to multiply the constants. Thus, we get the coefficient to be
$$
\begin{align}
\sum_{k=2}^n\sum_{j=1}^{k-1}jk
&=\sum_{k=2}^n\sum_{j=1}^{k-1}\binom{j}{1}k\\
&=\sum_{k=2}^n\binom{k}{2}k\\
&=\sum_{k=2}^n\binom{k}{2}((k-2)+2)\\
&=\sum_{k=2}^n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2527894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 4
} |
Solve $y''+4y=(1+\sin{x})^2$. My attempt: The null-solution is $A\cos{2x}+B\sin{2x}.$ Let's start by rewriting RHS by
$$(1+\sin{x})^2=1+2\sin{x}+\sin^2{x}=\frac{3}{2}+2\sin{x}-\frac{1}{2}\cos{x}.$$ So we can now look at too differential equations:
$$\left\{
\begin{array}{rcr}
y_1''+4y_1 & = & \frac{3}{2}+2\sin{x... | For a particular solution, you can solve for every term independently
$$y''+4y=1\to y=\frac 14.$$
$$y''+4y=\sin x\to y=\frac23\sin x$$ (by indeterminate coefficients).
Then $$y''+4y=\cos2x\to ???$$ (indeterminate coefficients don't seem to work).
It turns out that this RHS can be expressed as an instance of the homogen... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Elementary proof that $4$ never divides $n^2 - 3$ I would like to see a proof that for all integers $n$, $4$ never divides $n^2 - 3$. I have searched around and found some things about quadratic reciprocity, but I don't know anything about that. I am wondering if there is a more elementary proof.
For example, I managed... | It is obviously that if $n$ is even that $n^2-3$ is odd and so it is not divisible even by $2$.
Now suppose $n$ is odd. Among 4 consecutive integer exactly one is divisible by 4. So among $$n^2-3,\;\;\;\;n^2-2,\;\;\;\;n^2-1,\;\;\;\;n^2$$ exactly one is divisible by $4$. Since $n^2-1 = (n-1)(n+1)$ we see that $4|n^2-1$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Computing limit of $\sqrt{n^2+n}-\sqrt[4]{n^4+1}$ I have tried to solve this using conjugate multiplication, but I got stuck after factoring out $n^2$.
$\begin{align}
\lim_{n\rightarrow\infty}\dfrac{n^2+n-\sqrt{n^4+1}}{\sqrt{n^2+n}+\sqrt[4]{n^4+1}}
&=\lim_{n\rightarrow\infty}\dfrac{n(1+\dfrac{1}{n}-\sqrt{1+\dfrac{1}{n^... | One simple method is to try adding terms to give upper and lower bounds for the given roots. One may confirm (and not hard to find, either)
$$ \left( n + \frac{1}{2} - \frac{1}{8n} \right)^2 < n^2 + n < \left( n + \frac{1}{2} \right)^2 $$
$$ n^4 < n^4 + 1 < \left( n + \frac{1}{4n^3} \right)^4 $$
Together, we get
$$ ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
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Inverse Trigonometry System of Equations $$2\tan^{-1}\left(\sqrt{x-x^2}\right) = \tan^{-1}\left(x\right)\: +\, \tan^{-1}\left(1-x\right)$$
I have a feeling solution includes drawing triangles but cannot make the leap to get the solution
| We need $x(1-x)\ge0\iff x(x-1)\le0\iff0\le x\le1$
$\implies\dfrac{x+1-x}2\ge\sqrt{x(1-x)}$
Now use $$\arctan x+\arctan y= \arctan\frac{x+y}{1-xy}$$ if $xy<1$
See Inverse trigonometric function identity doubt: $\tan^{-1}x+\tan^{-1}y =-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right)$, when $x<0$, $y<0$, and $xy>1$
OR showing... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Expected number of games in a best of 7 series Assuming each team has a 0.5 probability of winning. Is there an easier way to do it other than bashing through the probabilities that the series runs 4,5,6,7 games.
| There are $2 \binom{3}{0}$ ways for the game to finish after $4$ rounds.
There are $2 \binom{4}{1}$ ways for the game to finish after $5$ rounds.
There are $2 \binom{5}{2}$ ways for the game to finish after $6$ rounds.
There are $2 \binom{6}{3}$ ways for the game to finish after $7$ rounds.
Quick sanity check
\begin{eq... | {
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"source": "stackexchange",
"question_score": "1",
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Improper integral of hyperbolic function I'm looking for an elementary derivation of the following formula:
$$\int_0^\infty \frac{\sinh(ax)}{e^{bx}-1}dx=\frac{1}{2a}-\frac{\pi}{2b}\cot\frac{a\pi}{b}$$
...only where $|a|\lt b$, of course, to ensure convergence. Does anyone know of any elementary ways of proving this i... | I give a real method that makes use of series and a bunch of special functions and their associated properties so perhaps this is not quite the elementary method you are after.
Let
$$I = \int^\infty_0 \frac{\sinh (ax)}{e^{bx} - 1} \, dx, \quad |a| < b.$$
Rewriting the hyperbolic sine function term appearing in the num... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to evaluate the integral $\int_0^{+\infty} \frac{\sin^4{x}}{x^4}dx$? As we know that $\int_0^{+\infty} \frac{\sin{x}}{x}dx=\pi/2$,but how to evaluate the integral $\int_0^{+\infty} \frac{\sin^4{x}}{x^4}dx$?
| Surely you know that$$\int\limits_0^{\infty}dx\,\frac {\sin^2x}{x^2}=\frac {\pi}2$$
which can be proven using integration by parts. Therefore, we take your integral by using integration by parts twice and a trigonometric identity to deduce$$\begin{align*}I & =-\frac {\sin^4x}{3x^3}\,\Biggr\rvert_{0}^{\infty}+\frac 43\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2535170",
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"source": "stackexchange",
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Product of Trigonometric function
The value of $$\prod^{10}_{r=1}\bigg(1+\tan r^\circ\bigg)\cdot \prod^{55}_{r=46}\bigg(1+\cot r^\circ\bigg)$$
Attempt: $\displaystyle \prod^{10}_{r=1}\bigg(1+\tan r^\circ\bigg)=(1+\tan 1^\circ)(1+\tan 9^\circ)\cdots \cdots (1+\tan 4^\circ)(1+\tan 6^\circ)\tan 5^\circ$
from $\tan(A+B) ... | $$ \prod^{10}_{r=1}\bigg(1+\tan r^\circ\bigg)\cdot \prod^{55}_{r=46}\bigg(1+\cot r^\circ\bigg) $$
$$ \prod^{10}_{r=1}\bigg( 1+\tan r^\circ \bigg)\cdot \prod^{10}_{r=1}\bigg(
1+\cot (45^\circ + r^\circ) \bigg)$$
$$ = \prod^{10}_{r=1}\bigg( (1+\tan r^\circ)(1+\cot (45^\circ + r^\circ)) \bigg) $$
$$ = \prod^{... | {
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"url": "https://math.stackexchange.com/questions/2537775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Taylor Polynomial $\ln(1-x)$ Given $f(x) = \ln(1-x)$ centered at $0$, with $n = 0,1,2$:
Why is it that the first derivative (for $n=1$): $$\frac{d}{dx}(p(x)) = -x$$ in a Taylor Polynomial? The answer that I got was $$\frac{d}{dx}(p(x)) = 1$$
I got the right answer for the second derivative $$\frac{d^2}{d^2x}(p(x)) = -... | \begin{align}
\int\frac{du}{u-1} = {} & \log\left| u-1 \right| + C \\[10pt]
= {} & \log(1-u) + C \text{ if $ u$ is near $1$} \\
& \text{since in that case, $u-1$ is negative.} \\[10pt]
\text{So } \int_0^x \frac{du}{u-1} & = \log(1-x) - \log(1-0) = \log(1-x).
\end{align}
Thus we have
\begin{align}
\log(1-x) & = \int_0^x... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I calculate the limit $\lim\limits_{x \to 7} \frac{\sqrt{x+2} - \sqrt[3]{x+20}}{\sqrt[4]{x+9} - 2}$ without L'Hospital's rule? I have a problem with calculation of the limit:
$$\lim\limits_{x \to 7} \frac{\sqrt{x+2} - \sqrt[3]{x+20}}{\sqrt[4]{x+9} - 2}$$
Is there a way to calculate it? How can I do it?
| Hint:
If $\sqrt{x+2}=a,\sqrt[3]{x+20}=b,\sqrt[4]{x+9}=c$
LCM$(2,3)=6$
$a^6-b^6=(a-b)(\cdots)$
Similarly $c^4-2^4=(c-2)\cdots$
| {
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"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Find the real solutions for the system: $ x^3+y^3=1$,$x^2y+2xy^2+y^3=2.$
Find the real solutions for the system:
$$\left\{
\begin{array}{l}
x^3+y^3=1\\
x^2y+2xy^2+y^3=2\\
\end{array}
\right.
$$
From a book with exercises for math contests. The solutions provided are: $(x,y)=(\dfrac{1}{\sqrt[3]{2}},\dfrac{1}{\sqr... | Multiply the first equation by $2$ and then set the two left sides equal:
$$2x^3+2y^3 = x^2y+2xy^2+y^3.$$
This is a homogeneous equation of degree $3$, so divide through by $x^3$:
$$2+2\frac{y^3}{x^3} = \frac{y}{x}+2\frac{y^2}{x^2}+\frac{y^3}{x^3}.$$
Substitute $u=y/x$:
$$2+2u^2=u+2u^2+u^3$$
$$u^3-2u^2-u+2=0$$
$$(u^2-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2544262",
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"source": "stackexchange",
"question_score": "2",
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Simple solution to this algebraic equation? Consider this system of equations with two variables $x,y$ and all positive parameters
$$\frac{2(x-w_1)}{a_1}=\frac{x}{\sqrt{x^2+y^2}}\,,\quad\frac{2(y-w_2)}{a_2}=\frac{y}{\sqrt{x^2+y^2}}.$$
It at most amounts to an algebraic equation of degree 4, which is always solvable in ... | It follows from
\begin{equation}
\frac{(x-w_1)^2}{a_1^2} + \frac{(y-w_2)^2}{a_2^2} = \frac{1}{2^2}
\end{equation}
that
the solutions are lying on the ellipses.
Thus
\begin{equation}
y = w_2 \pm \sqrt{\frac{a_2^2}{2^2} - \frac{a_2^2(x-w_1)^2}{a_1^2}}
\end{equation}
Then plugging this into the below
\begin{equation}
\fr... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exercise on elementary number theory Let $a,b,c,d$ integers, $c\not=0$ such that $ad-bc=1$ and $c\equiv 0 \pmod p$ for some prime $p>3$.
Show that if $a+d=\pm1$ then $p\equiv 1\pmod 3$
I don't know how to approach this problem because when I take the expression $ad-bc$ modulo $p$ we have that either $d-d^2\equiv 1\pm... | Let's consider:
$ad-bc\equiv1 \pmod p\implies ad=1-bc \equiv 1\pmod p$
$(a+d)^2 \equiv a^2+d^2+2ad\equiv a^2+d^2+2\equiv1\pmod p$
$\implies a^2+d^2+1\equiv 0\pmod p$
$\implies (a^2+d^2+1)^2\equiv 0\pmod p$
Since:
$a+d=\pm1 \implies a\cdot d=(k+1)\cdot(-k)=-k^2-k$
Consider the following table $\pmod 3$
$$\begin{array}{ ... | {
"language": "en",
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"source": "stackexchange",
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Prove that the roots of $ax^2+bx+c=0$, $a\neq 0$ and $a,b,c\in R$ will be real if $a(a+7b+49c)+c(a-b+c)<0$
Prove that the roots of $ax^2+bx+c=0$, $a\neq 0$ and $a,b,c\in \mathbb{R}$ will be real if $$a(a+7b+49c)+c(a-b+c)<0$$
My Attempt:
Given
\begin{align}
a(a+7b+49c)+c(a-b+c) &< 0 \\
49a \left( \dfrac {a}{49} + ... | We can assume that $a=1$ and let $f(x) =x^2+bx+c$.
If $c< 0$ then the graph of $f$ cuts the $y-$ axis under the $x-$ axis, so it must have real roots.
If $c=0$ then 0ne root is $0$ and the second is $-b$.
If $c>0$, since we have $$49f({1\over 7})+cf(-1)=0$$
we have 2 possibilities.
a) If $f(-1)=f(1/7)=0$ we are don... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Closed form of $I(t) = \int_0^{\pi/2}\frac{\cos xdx}{\sqrt{\sin^2 x+ t\cos^2 x}}$ What is the closed form of
$$I(t) = \int_0^{\pi/2}\frac{\cos xdx}{\sqrt{\sin^2 x+ t\cos^2 x}}$$
I tired the change of variables $$\int_a^bf(x) dx= \int_a^bf(a+b-x) dx$$
but is was not fruitful.
| For $t > 1$
$$I(t) = \int_0^{\pi/2}\frac{\cos xdx}{\sqrt{\sin^2 x+ t\cos^2 x}} = \int_0^{1}\frac{d(\sin x)}{\sqrt{\sin^2 x(1- t)+ t }} = \dfrac{\arcsin\left(\frac{\sqrt{t-1}}{\sqrt{t}}\right)}{\sqrt{t-1}}$$.
For $0 < t < 1$, let $ u = \sin x$
$$I(t) = \int_0^1 \dfrac{du}{\sqrt{u^2(1 - t) + t}} = \dfrac{1}{\sqrt t}\int... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Estimate from below of the sine (and from above of cosine) I'm trying to do the following exercise with no success. I'm asked to prove that
$$\sin(x) \ge x-\frac{x^3}{2}\,, \qquad \forall x\in [0,1]$$
By using Taylor's expansion, it's basically immediate that one has the better estimate
$$\sin(x) \ge x-\frac{x^3}{6}\,,... | A geometric proof is as follows.
Outline:
*
*Show $\cos x > 1-\frac{1}{2}x^2.$
*Show that $\tan x> x.$
From there, you quickly see that $\sin x>x\cos x>x-x^3/2.$
We have that $\sqrt{(1-\cos x)^2+\sin^2 x}$ is the length of segment from $(1,0)$ to $(\cos x,\sin x)$, which is $\leq x$, since the arc along the circl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2555669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Find the limit of a series of fractions starting with $\frac{1}{2}, \, \frac{1/2}{3/4}, \, \frac{\frac{1}{2}/\frac{3}{4}}{\frac{5}{6}/\frac{7}{8}}$ Problem
Let $a_{0}(n) = \frac{2n-1}{2n}$ and $a_{k+1}(n) = \frac{a_{k}(n)}{a_{k}(n+2^k)}$ for $k \geq 0.$
The first several terms in the series $a_k(1)$ for $k \geq 0$ are:... | Let $f_0(z) = z$ and $f_{n+1}(z) = f_n(z) / f_n(z+2^n)$
One can show that when $z \to \infty$, the rational fractions $f_n$ for $n \ge 1$ have asymptotic developments at infinity that converge for $|z| > n$, such that
$f_n(z) = 1 + O(z^{-n})$ and $f_n'(z) = O(z^{-n-1})$
Call $s(k) = +1,-1,-1,+1,\cdots$ the Thue-Morse ... | {
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"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Simplify expression $\frac{2\cos(x)+1}{4\cos(x/2+π/6)}$ How to simplify the following expression: $$\frac{2\cos(x)+1}{4\cos\left(\frac x2+\fracπ6\right)}$$
I got to: $ \dfrac{2\cos(x)+1}{4\cos\left(\dfrac x2\right)\cdot \dfrac{\sqrt3}2-\sin(x) \cdot \frac 12}$
| Using half-angle formula: $\cos\dfrac A2=\pm\sqrt{\dfrac{1+\cos A}2}$ with $A=x+\dfrac\pi3$ to give $$\cos\left(\frac x2 + \frac \pi6 \right)=\pm\sqrt{\dfrac{1+\cos \left(x + \frac \pi3 \right)}2}=\pm\sqrt{\dfrac{1+\frac{\sqrt3}2\cos x-\frac12 \sin x }2}$$ so $$\cos\left(\frac x2 + \frac \pi6 \right)=\pm\frac{\sqrt{1+\... | {
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"url": "https://math.stackexchange.com/questions/2558565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Non-empty limit set for the dynamical system : $x_1' = x_1 + 2x_2 - 2x_1(x_1^2 + x_2^2)^2, \space x_2' = 4x_1 + 3x_2 - 3x_2(x_1^2 + x_2^2)^2 $
Using the Lyapunov Function $V=\frac{1}{2}x_1^2 + \frac{1}{2}x_2^2$ , prove that the omega(ω)-limit set is non-empty for any initial value given for the dynamical system :
$$... | The Cauchy's inequality can be rewriten in the form
$$(x_1\cdot 1+x_2\cdot 1)^2\le (x_1^2+x_2^2)\cdot (1^2+1^2)$$
or $$(x_1+x_2)^2\le 2(x_1^2+x_2^2).$$
This implies that
$$
\dot V\leq 3(x_1+x_2)^2 - 2(x_1^2 + x_2^2)^3\le
6(x_1^2+x_2^2)-2(x_1^2+x_2^2)^3
$$
It is easy to check that $\forall (x_1,x_2):\; x_1^2+x_2^2>\s... | {
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"source": "stackexchange",
"question_score": "3",
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Use Viete's relations to prove the roots of the equation $x^3+ax+b=0$ satisfy $(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2=-4a^3-27b^2$ Use Viete's relations to prove that the roots $x_1$, $x_2$, and $x_3$ of the equation $x^3+ax+b=0$ satisfy the identity $(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2=-4a^3-27b^2$.
I know that viete's relat... | Let
$$f(x)=x^3+ax+b=(x-x_1)(x-x_2)(x-x_3)$$
so that
$$f'(x)=3x^2+a=(x-x_1)(x-x_2)+(x-x_1)(x-x_3)+(x-x_2)(x-x_3)$$
This allows us to get the equation
$$ f'(x_1)f'(x_2)f'(x_3)=(3x_1^2+a)(3x_2^2+a)(3x_3^2+a)=-(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 $$
We try to evaluate
$$
\begin{align}
(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 &=
-(3... | {
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"url": "https://math.stackexchange.com/questions/2564158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find all real solutions for the system: $x^3=y+y^5$, $y^5=z+z^7$, $z^7=x+x^3.$
Given: $$\left\{
\begin{array}{l}
x^3=y+y^5\\
y^5=z+z^7\\
z^7=x+x^3
\end{array}
\right.
$$
Find: all real solutions for the system.
From a book on preparation for math contests. The answer states there is just one solution. My problem... | Let $x>0$.
Hence, $z^7=x(1+x^2)>0$, which gives $z>0$.
Also, $y^5=z(1+z^6)>0$, which gives $y>0$.
But summing of all equations gives $x+y+z=0$, which is a contradiction.
By the same way we can get a contradiction for $x<0$.
Thus, $x=0$ and from here we obtain $x=y=z=0.$
| {
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"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Solve $\sqrt{x^2+1}-\sqrt{4x^4-4x^2+2}=2x^3-x-1$
Could you please help me solve for $x$ in
$$\sqrt{x^2+1}-\sqrt{4x^4-4x^2+2}=2x^3-x-1.$$
I tried this way. But I could not solve further. Please help me.
$$(\sqrt{x^2+1}-\sqrt{4x^4-4x^2+2})^2=(2x^3-x-1)^2$$
| As $x$ increases:
$\sqrt{4x^4-4x^2+2} = \sqrt{(2x^2-1)^2+1}$ decreases if $x \in [0,\frac12\sqrt2]$ and increases if $x \in [\frac12\sqrt2,-\infty)$.
Symmetrically $\sqrt{4x^4-4x^2+2}$ decreases if $x \in (-\infty,-\frac12\sqrt2]$ and increases if $x \in [-\frac12\sqrt2,0]$.
Therefore $\sqrt{4x^4-4x^2+2} \le \sqrt{... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Get closed form from a complicated closed-form generating function I have a closed form for a generating function:
$A(x)=\frac{x(x-1)(x+1)^3(x^3-x-1)}{(x^3+x^2-1)^2}$
The coefficient of $x^n$ in the above represents $a_n$ (the $n^{th}$ term of a sequence). I want a closed form for the sequence $a_n$. For example, the e... | Hint: Though not a closed form we can use the binomial series expansion to derive an alternate explicit expression for $[x^n]A(x)$, the coefficient of $x^n$ of $A(x)$.
At first we write $A(x)$ with increasing powers of the denominator $1-x^2-x^3$. We obtain using the Euclidian algorithm for polynomials:
\begin{align... | {
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"answer_id": 1
} |
Evaluating $\lim_{x \to 1^{-}} \prod_{n=0}^{\infty} \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$ If $$\lim_{x \to 1^{-}} \prod_{n=0}^{\infty} \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}=l$$ the question is to find out the integer part of the number $1/l$
I tried bringing the expression into telescopic series but failed t... | Let's consider the logarithm of this product. The partial sum of the resulting series is
\begin{align}
\sum^n_{k=0}x^k\ln\frac{1+x^{k+1}}{1+x^k}&=\sum^n_{k=0}x^k\ln(1+x^{k+1})-\sum^n_{k=0}x^k\ln(1+x^k)
\\&=\sum^n_{k=0}x^k\ln(1+x^{k+1})-\sum^{n+1}_{k=1}x^k\ln(1+x^k)-\ln2+x^{n+1}\ln(1+x^{n+1})
\\&=\sum^n_{k=0}x^k\ln(1+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to prove $\prod_{k=1}^{\infty}{\frac{p_k^2+1}{p_k^2-1}}=\frac{5}{2}$? I came across the following formula due to Ramanujan:
$$\prod_{k=1}^{\infty}{\frac{p_k^2+1}{p_k^2-1}}=\frac{5}{2}.$$
Can someone show me what the proof of this looks like, or point me to a reference (in English)?
| By Euler's product, for any $s>1$ we have
$$ \zeta(s)=\sum_{n\geq 1}\frac{1}{n^s} = \prod_{p}\left(1-\frac{1}{p^s}\right)^{-1} \tag{A}$$
hence
$$ \prod_{p}\frac{p^2+1}{p^2-1} = \prod_p\frac{1-\frac{1}{p^4}}{\left(1-\frac{1}{p^2}\right)^2} = \frac{\zeta(2)^2}{\zeta(4)} = \frac{90}{36} = \frac{5}{2}.\tag{B}$$
Similarly, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 1,
"answer_id": 0
} |
Finding Eigenvectors [Confused] I would like to find the eigenvalues for the matrix
$$\begin{pmatrix}
2 & 3-3i \\
3+3i & 5
\end{pmatrix}$$
I find that the eigenvalues are $8$ and $-1$.
For eigenvalue of $8$ I get
$$
\begin{pmatrix}
-2 & 1-1i \\
1+1i & -1
\end{pmatrix}=0
$$
and I get the equation
$$-2x + (1-1i)y = 0\\... | First the eigenvalues
$\begin{align}\det(A-tI)&=\begin{vmatrix}2-t & 3-3i\\ 3+3i & 5-t\end{vmatrix}=(2-t)(5-t)-(3+3i)\overline{(3+3i)}=10-7t+t^2-18\\\\&=t^2-7t-8=(t+1)(t-8)\end{align}$
So you are correct, $\operatorname{Sp}(A)=\{-1,8\}$.
Now for the eigenvectors you want to solve two systems for $v=(x,y)^T$:
*
*$Av=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$\lim_{x\to 0} \left(\frac{1}{1-\cos x}-\frac{2}{x^2}\right)$ Find the limits :
$$\lim_{x\to 0} \left(\frac{1}{1-\cos x}-\frac{2}{x^2}\right)$$
My Try :
$$\lim_{x\to 0} \left(\frac{1}{1-\cos x}-\frac{2}{x^2}\right)=\lim_{x\to 0}\frac{x^2-2(1-\cos x)}{x^2(1-\cos x)}$$
Now what do I do ?
| $$\lim_{x \to 0}\frac{1}{1-(1-\frac{x^2}{2!}+O(x^4))} - \frac{2}{x^2} = \frac{2}{x^2}-\frac{2}{x^2} = 0$$(higher order terms neglected as the become very small as $x \to0$)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
find the range of the function : $y=(3\sin 2x-4\cos 2x)^2-5$ find the range of the function :
$$y=(3\sin 2x-4\cos 2x)^2-5$$
My try :
$$y=9\sin^22x+16\cos^22x-24\sin 2x\cos 2x-5\\y=9+7\cos^22x-12\sin4x-5$$
now what do I do؟
| You can write $$3\sin 2x-4\cos 2x =5\sin(2x+\varphi)$$
for some constant angle $\varphi$,
so $$ y = 25\sin^2(2x+\varphi)-5$$
so $y_{\max} = 20$ and $y_{\min} = -5$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Find $x$ given $\sqrt{x+14-8\sqrt{x-2}}$ + $\sqrt{x+23-10\sqrt{x-2}} = 3$ If $\sqrt{x+14-8\sqrt{x-2}}$ + $\sqrt{x+23-10\sqrt{x-2}} = 3$, then what is the value of $x$?
Is there an easy way to solve such equation, instead of squaring on both sides and replacing $\sqrt{x-2}$ with a different variable?
| It's not so bad; after you substitute $y=\sqrt{x-2}$ you will have
$$\sqrt{y^2 - 8y + 16 } + \sqrt{y^2 - 10y+ 25 } = 3$$
and you should be able to take it from here without any squaring of both sides.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2574220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Proving whether the series $\sum_{n=1}^\infty \frac{(-1)^n}{n-(-1)^n}$ converges. I've updated my proof to be complete now, edited for proof-verification!
We know that for the partial sums with even an uneven terms, the following holds:
$S_{2N}=\sum_{n=1}^{2N} \frac{(-1)^n}{n-(-1)^n} = -\frac{1}{2} + \frac{1}{1} -\fra... | Another way is to observe that
$$\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n-(-1)^{n}}=\sum_{n=1}^{\infty}\Bigg[\frac{(-1)^{n}}{n-(-1)^{n}}-\frac{(-1)^{n}}{n}\Bigg]+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}=$$
$$=\sum_{n=1}^{\infty}\frac{1}{n^{2}-(-1)^{n}n}+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}$$
Now the left series converges by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2575967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
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Proving formally that a set is subspace. I want to prove formally that $U = (x,y,z\in \mathbb{R^3})$
$| \det\begin{bmatrix}a_{11}&a_{12} & x\\
a_{21}&a_{22}&y\\ a_{31}&a_{32}&z \end{bmatrix} = 0$ is a subspace.
I know this is true but the technique I learnt, I think I can't apply it here.
To prove that they are cl... | Useful observation:
$$\det\begin{bmatrix}a_{11}&a_{12} & x\\
a_{21}&a_{22}&y\\ a_{31}&a_{32}&z \end{bmatrix} = 0 \iff \left\{\begin{pmatrix}a_{11} \\ a_{21} \\a_{31} \end{pmatrix}, \begin{pmatrix}a_{12} \\ a_{22} \\a_{32} \end{pmatrix}, \begin{pmatrix}x \\ y \\z \end{pmatrix}\right\} \text{ is linearly dependent}$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
2nd solution of $\cos x \cos 2x\cos 3x= \frac 1 4 $
$\cos x \cos 2x\cos 3x= \dfrac 1 4 $
Attempt explained:
$(2\cos x \cos 3x)\cos 2x = \frac1 2 $
$(\cos 4x +\cos 2x )\cos 2x = \frac 1 2 \\\cos ^2y + \cos y (2\cos^2y- 1)= \frac1 2 \\ $
(Let, y = 2x)
$\implies 4\cos^3 y+2\cos^2y- 2\cos y-1=0$
I solved this equation us... | All my answers are correct. However, the last answer had to be obtained using a different form. I had used the general solution of $\cos^2 x = \cos ^2 \alpha$ but they expected me to use the general form of $\cos x = 0$
"Correction":
$\cos^2 y = \frac 1 2 \implies \cos 2y = \cos 0 \implies 2y = (2n+1)\frac\pi2 \implie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Inverse of Laplace operator Let $\ L=\frac{-d^2}{dx^2}$ defined on $\ H^2(]0,1[) ∩ H^1_0(]0,1[)$ \ $\ L$ is the laplacian operator in one dimension \ how can we express the inverse of $\ L $
| The Green's function might need a slight modification.
\begin{align*}
G\left(x,y\right) & =\frac{2}{\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}}\sin\left(n\pi x\right)\sin\left(n\pi y\right)\\
& =\begin{cases}
\left(1-y\right)x & 0\leq x\leq y\\
y\left(1-x\right) & y\leq x\leq1
\end{cases}.
\end{align*}
In details:
Le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find the Bezout coefficients Find the Bezout coefficients for $a(x)$ and $b(x)$:
$a(x)=3x^4-4x^3-11x^2+4x+9, b(x)=3x^3+5x^2+x-1$
I find the greatest common divisor:
1) $\frac{(3x^4-4x^3-11x^2+4x+9)}{(3x^3+5x^2+x-1)} = x-3$. Remainder of the division: 3x^2+8x+6
2) $\frac{(3x^3+5x^2+x-1)}{(3x^2+8x+6)} = x-1$. Remainder o... | I got
$$ \left( 3 x^{4} - 4 x^{3} - 11 x^{2} + 4 x + 9 \right) $$
$$ \left( 3 x^{3} + 5 x^{2} + x - 1 \right) $$
$$ \left( 3 x^{4} - 4 x^{3} - 11 x^{2} + 4 x + 9 \right) = \left( 3 x^{3} + 5 x^{2} + x - 1 \right) \cdot \color{magenta}{ \left( x - 3 \right) } + \left( 3 x^{2} + 8 x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Example of multivalued function (includes pairwise comparisons) that attains maximum when values form evenly spaced vector Suppose, that I have 4 points $x,y,z,w$ (positions on the x-axis), such that $x\le y\le z\le w$.
I find pairwise distances between them: $y-x,z-x,w-x,z-y,w-y,w-z$. Distances should be less or equal... | Let $\,g\,$ be a strictly concave and strictly increasing function on $\,[\,0,k\,]\,$ such that $\,g(0)=0\,$.
Define $\,f : \left\{\, (x,y,z,w) \in [\,1,k+1\,]^4 \;\mid\; x \le y \le z \le w \,\right\} \to \mathbb{R}\,$ as:
$$
\begin{align}
f(x,y,z,w) \;&=\; 2g(y-x)+g(z-y)+2g(w-z) \\[3px]
&\quad +\;2 g\left(\frac{z-x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\max a^3+b^3+c^3+4abc$ sub $0\leq a,b,c \le 3/2$ and $a+b+c=3$ Let $S$ be the set of $(a,b,c) \in \mathbb{R}^3$ such that $0\leq a,b,c \leq \frac{3}{2}$ and $a+b+c=3$. Find
$$
\max_{(a,b,c) \in S} a^3+b^3+c^3+4abc.
$$
| Let us start with a triple $(a, b, c)$. Let us try to "tune" the triple and see if we can come out with a candidate $(a', b', c')$ with a larger $a'^3 + b'^3 + c'^3 + 4a'b'c'$. Heuristic tells us that $(a', b', c') = (a, \frac{b+c}{2}, \frac{b+c}{2})$ worth some consideration.
Let $A = a^3 + b^3 + c^3 + 4abc$ and $B = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Integral of rational function - which contour to use? Evaluate : $$\int_{-\infty}^{+\infty} \frac {x}{(x^2+2x+2)(x^2+4)}$$
I found that the integrand can be extended to a function on a complex plane has simple poles at $\pm 2i$ and $-1\pm i$. Now I want to compute the integral by contour integration but I am unable to... | Hint: Use upper half plane as contour and fraction decomposition
$$\dfrac{z}{(z^2+2z+2)(z^2+4)}=\dfrac{1}{10}\frac{z-2}{z^2+2z+2}-\dfrac{1}{10}\frac{z-4}{z^2+4}$$
then
$$\dfrac{1}{10}\int_C\frac{z-2}{z^2+2z+2}-\frac{z-4}{z^2+4}dz=\dfrac{2\pi i}{10}\left(\operatorname*{Res}_{z=i-1}\frac{z-2}{z^2+2z+2}-\operatorname*{Res... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Prove that $\sqrt{s(s-a)(s-b)(s-c)}=A$ In 50 AD, the Heron of Alexandria came up with the well-known formula, that, given the three side lengths of a triangle (or even two and an angle, thanks to trigonometry) you can get the area of said triangle by using this formula:
$$
\text{if } s=\frac{a+b+c}{2},\\
\text{then} A=... | Take a look here for the proof: Heron's formula
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2582044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Circumcircle and Square Let $P$ be a point on the circumcircle of square $ABCD$. Find all integers
$n > 0$ such that the sum
$$S^n(P) = PA^n + PB^n + PC^n + PD^n$$
is constant with respect to point $P$.
| Say $x$ is a side of $ABCD$ and $a=PA$, $b=PB$...
If $n=1$ then the sum is not constant:
Say $P$ is on smaller arc $BC$. Then by Ptolomey theorem for $ABPC$ we have $$ax =bx\sqrt{2}+cx$$ and by Ptolomey theorem for $DBPC$ we have $$dx= bx+cx\sqrt{2}$$
so $$a+d = (b+c)(1+\sqrt{2}) \Longrightarrow a+b+c+d = (2+\sqrt{2})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2582134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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product of terms taken $3$ at a time in polynomial expression
Finding product of terms taken $3$ at a time in $\displaystyle \prod^{100}_{r=1}(x+r)$
Try:
$$\displaystyle \prod^{100}_{r=1}(x+r)=x^{100}+(1+2+3+\cdots +100)x^{99}+(1\cdot 2+1\cdot 3+\cdots+100\cdot 99)x^{98}+(1\cdot 2\cdot 3+2\cdot 3 \cdot 4+\cdot\cdot... | With some manipulation (using inclusion exclusion principle) you can write the sum as:
$$\left(\sum_{i=1}^{100} i\right)^3 = \binom{3}{1}\left(\sum_{i=1}^{100} i^2\right)\left(\sum_{i=1}^{100} i\right) - 2\left(\sum_{i=1}^{100}i^3\right) + \binom{3}{1} \binom{2}{1}S $$
Where $S$ is the required sum.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2582647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Taylor limits with sine I'm having troubles calculating these two limits (I prefer to write a sigle question including both of them, instead of two different ones).
This one
$$\\ \lim_{x\rightarrow 0} \frac{ x-\sin^2(\sqrt x)-\sin^2(x)} {x^2} $$ I tried expanding with Taylors at different orders but the... | Second limit
Note that:
$$(1-x)^{-1}=1+x+x^2+x^3+o(x^3)$$
$$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+o(x^3)$$
$$({(1-x)^{-1}} +e^x)^2=\left(1+x+x^2+x^3+1+x+\frac{x^2}{2}+\frac{x^3}{6}+o(x^3)\right)^2=\left(2+2x+\frac{3x^2}{2}+\frac{7x^3}{6}+o(x^3)\right)^2=4+4x^2+8x+6x^2+6x^3+\frac{14x^3}{3}+o(x^3)=4+8x+10x^2+\frac{32x^3}{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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How is that $A^n A\neq A A^n$? How is that $A^n A\neq A A^n$? Where $A$ is a $n\times n$ matrix whose elements belong to a set with ring structure(it may not be commutative).
$A^nA$ can be expressed as $(AA\cdots AA)A$, which, by associativity of matrix multiplication it means that it is the same as $A(AA\cdots AA)$
Su... | Do a little more algebra and expand out the terms. For example,
$$ (AA^2)_{1,1} = a(a^2+bc) + b(ca + dc) = a^3 + abc + bca + bdc $$
and
$$ (A^2A)_{1,1} = (a^2+bc)a + (ab + bd)c = a^3 + bca + abc + bdc. $$
Since addition is commutative, these two terms are equal. The remaining entries are similar.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to evaluate $1 - \frac{\binom{n^2}{1}}{\binom{n+1}{1}} + \frac{\binom{n^2}{2}}{\binom{n+2}{2}} - \frac{\binom{n^2}{3}}{\binom{n+3}{3}} + ..$ How to evaluate $1 - \frac{\binom{n^2}{1}}{\binom{n+1}{1}} + \frac{\binom{n^2}{2}}{\binom{n+2}{2} } - \frac{\binom{n^2}{3}}{\binom{n+3}{3}} + \frac{\binom{n^2}{4}}{\binom{n+4}... |
A variation. We obtain
\begin{align*}
\color{blue}{\sum_{j=0}^{n^2}}&\color{blue}{(-1)^j\binom{n^2}{j}\binom{n+j}{j}^{-1}}\\
&=\sum_{j=0}^{n^2}\binom{n^2}{j}\binom{-n-1}{j}^{-1}\tag{1}\\
&=\sum_{j=0}^{n^2}\binom{n^2}{j}(-n)\int_{0}^1t^j(1-t)^{-n-1-j}\,dt\tag{2}\\
&=(-n)\int_{0}^1(1-t)^{-n-1}\sum_{j=0}^{n^2}\binom{n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
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$\lim_{n\rightarrow \infty}\left[\frac{\left(1+\frac{1}{n^2}\right)\cdot \cdots\cdots \left(1+\frac{n}{n^2}\right)}{\sqrt{e}}\right]^n$
$$\lim_{n\rightarrow \infty}\Bigg[\frac{\bigg(1+\frac{1}{n^2}\bigg)\bigg(1+\frac{2}{n^2}\bigg)\cdots\cdots \bigg(1+\frac{n}{n^2}\bigg)}{\sqrt{e}}\Bigg]^n$$
Try: $$y=\lim_{n\rightarro... | We have $$\log{\left ( 1+\frac{r}{n^2} \right )}=\frac{r}{n^2}-\frac12\frac{r^2}{n^4}+O\left(\frac{r^3}{n^6}\right),$$ that means
$$n\sum^n_ {r=1}\log{\left ( 1+\frac{r}{n^2} \right )}=\frac{n(n+1)}2\frac1n-\frac12\frac{n(n+1)(2n+1)}6\frac1{n^3}+O(n^4)\frac1{n^5},$$ so we get
$$n\sum^n_ {r=1}\log{\left ( 1+\frac{r}{n^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
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Point of intersection when only direction ratios are given I am starting out with 3D Geometry. In one of the test booklets, I found a question for which I have no idea where and how to start from.
If a line with direction ratio $2:2:1$ intersects the line $\frac{x-7}{3}$ = $\frac{y-5}{2}$ = $\frac{z-3}{2}$ and $\frac... | Let $a$ be the common value of $$\frac{x-7}{3}=\frac{y-5}{2}=\frac{z-3}{2}=a$$
We can re-write these equations under the following equivalent parametric form:
$$\tag{1}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}7+3a\\5+2a\\3+2a\end{pmatrix}$$
In the same way, the generic point of the second straight line is : ... | {
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"url": "https://math.stackexchange.com/questions/2594035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that $\frac{a+b}{1+a+b} \leq \frac{a}{1+a} + \frac{b}{1+b}$ for non-negative $a,b$? If $a, b$ are non-negative real numbers, prove that
$$
\frac{a+b}{1+a+b} \leq \frac{a}{1+a} + \frac{b}{1+b}
$$
I am trying to prove this result. To that end I added $ab$ to both denominator and numerator as we know
$$
\frac... | I guess you're almost there...
If $a\ge 0$, then $1+a\ge 1$ and so $(1+a)(1+b)\ge 1+b$. This gives you
$$\frac b {(1+a)(1+b)}\le \frac b {1+b}$$
and the result you're looking for follows.
| {
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"url": "https://math.stackexchange.com/questions/2595966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
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$2^{x-3} + \frac {15}{2^{3-x}} = 256$ $$2^{x-3} + \frac {15}{2^{3-x}} = 256$$
*
*Find the unknown $x$.
My attempt:
We know that $x^y . x^b = x^{y+b}$.
$$2^x . 2^{-3} + 15. 2^{-3+x} = 2^8$$
and
$$2^x . 2^{-3} + 15. 2^{-3} . 2^x = 2^8$$
From here, we get
$$2^x + 15 = 2^8$$
However, I'm stuck at here and waiting f... | writing $$\frac{2^x}{8}+\frac{15}{8}2^x=2^8$$ so
$$2^x\left(\frac{1}{8}+\frac{15}{8}\right)=2^8$$
Can you finish?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Summation. What does is evaluate to? What is $\sum_{n=1}^{\infty} \frac{a_{n}}{4^{n+1}}$ if $a_{n+2}=a_{n+1}+a_{n}$ and $a_{1}=a_{2}=1$?
| $a_{n+2}=a_{n+1}+a_{n}$
with
$a_1=a_2 = 1$.
Let
$f(x)
=\sum_{n=1}^{\infty} a_nx^n
$.
$xf(x)
=\sum_{n=1}^{\infty} a_nx^{n+1}
=\sum_{n=2}^{\infty} a_{n-1}x^{n}
$
and
$x^2f(x)
=\sum_{n=1}^{\infty} a_nx^{n+2}
=\sum_{n=3}^{\infty} a_{n-2}x^{n}
$
so
$\begin{array}\\
xf(x)+x^2f(x)
&=\sum_{n=2}^{\infty} a_{n-1}x^{n}+\sum_{n=3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Find the maximum of the expression Let $a,b,c$ be real positive numbers so that $abc=1$. Find the maximum value that the following expression can attain:
$$\frac{a}{a^8+1}+\frac{b}{b^8+1}+\frac{c}{c^8+1}$$
My try:
I first though on apply a variable change so that $a=\frac{x}{y}$, $b= \frac{y}{z}$ and $c=\frac{z}{x}$. T... | We'll prove that $$\frac{a}{a^8+1}\leq\frac{3(a^6+1)}{4(a^{12}+a^6+1)}.$$
Indeed, we need to prove that
$$\frac{1}{a^4+\frac{1}{a^4}}\leq\frac{3\left(a^3+\frac{1}{a^3}\right)}{4\left(a^6+\frac{1}{a^6}+1\right)}.$$
Let $a+\frac{1}{a}=2t$.
Thus, by AM-GM $t\geq1$ and we need to prove that:
$$\frac{1}{16t^4-16t^2+2}\leq\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Definite integral of a rational fraction Can I find the value of $$\int_3^{\infty}\frac{x-1}{(x^2-2x-3)^2}dx$$ by just factoring the fraction?
I tried to wrote:
$$\frac{x-1}{(x^2-2x-3)^2}=\frac{x-1}{(x^2-2x+1-4)^2}=\frac{x-1}{[(x-1)^2-2^2]^2}=\frac{x-1}{(x+1)^2\cdot(x-3)^2}$$ but didn't work out. Any ideas?
| Use partial fraction decomposition, that is, find $A,B,C,D\in\mathbb{R}$ such that$$\frac{x-1}{(x+1)^2(x-3)^2}=\frac A{x+1}+\frac B{(x+1)^2}+\frac C{x-3}+\frac D{(x-3)^2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2601228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Find the angle between the two tangents drawn from the point $(1,2)$ to the ellipse $x^2+2y^2=3$.
Find the angle between the two tangents drawn from the point $(1,2)$ to the ellipse $x^2+2y^2=3$.
The given ellipse is $\dfrac{x^2}{3}+\dfrac{y^2}{\frac{3}{2}}=1$
Any point on the ellipse is given by $(a\cos \theta,b\sin... | If $y=mx+n$ is a tangent then $$n^2=a^2m^2+b^2$$ or
$$n^2=3m^2+\frac{3}{2}.$$
Also, we have $2=m+n$ and we got the following equation on slopes:
$$(2-m)^2=3m^2+\frac{3}{2}.$$
After this use $$\tan\alpha=\left|\frac{m_1-m_2}{1+m_1m_2}\right|.$$
I got $$\alpha=\arctan12.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2602561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Proving a small inequality I am given that $a^3+b^3+c^3=3$ where a, b, c are positive numbers and I need to prove that
$$\frac {3(ab+bc+ac)+a^3c^2+b^3a^2+c^3b^2}{(a+b)(b+c)(a+c)}\ge \frac {3}{2}$$
At first it seems to me that the inequality might be wrong. I have tried using the Cauchy Schwarz , AM GM and some algebrai... | We need to prove that
$$\sum_{cyc}\frac{3ab+a^3c^2}{\prod\limits_{cyc}(a+b)}\geq\frac{3}{2}$$ or
$$\sum_{cyc}\frac{(a^3+b^3+c^3)ab+a^3c^2}{\prod\limits_{cyc}(a+b)}\geq\frac{3}{2}$$ or
$$\sum_{cyc}\frac{a^4b+a^4c+a^3bc+a^3c^2}{\prod\limits_{cyc}(a+b)}\geq\frac{3}{2}$$ or
$$\sum_{cyc}\frac{a^3(ab+ac+bc+c^2)}{\prod\limits... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2604465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Remainder of $22!$ upon division with $23$? I couldn't solve the problem, but I came to know the answer is $22$.
Then I tried to check the numbers in factorial will be cancelled by their modulo inverses w.r.t $23$. But they didn't.
\begin{array}{|c|c|} \hline \text{Number} & \text{Modulo Inverse w.r.t 23} \\ \hline
2 &... | Your (multiplicative) inverses should be $$2\times 12=3\times 8=4\times 6=24\equiv 1 \bmod 23$$
$$5\times 14=7\times 10=70\equiv 1\bmod 23$$ $$9\times 18 \equiv 1 \bmod 23 $$
$$11\times21=231\equiv 1\bmod 23$$$$13\times 16=208\equiv 1 \bmod 23$$$$15\times 20=299\equiv 1 \bmod 23$$$$17\times 19=323\equiv 1 \bmod 23$$
Yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2604626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why the approximate solution of $\frac{1}{{\sqrt {2\pi } }}\frac{n}{x}{e^{ - \frac{{{x^2}}}{2}}} = c$ is $\sqrt {2\log n} $ when $n$ is large I find in a book that when $n$ is large, the approximate solution of $\frac{1}{{\sqrt {2\pi } }}\frac{n}{x}{e^{ - \frac{{{x^2}}}{2}}} = c$, denoted by $x(n,c)$, is about
$x(n,c... | The initial equation can be written
$$e^{-x^2/2-\log x}=\frac{\sqrt{2\pi}c}n,$$
and taking the cologarithm,
$$\frac{x^2}2+\log x=\log n-\log\sqrt{2\pi}c.$$
For very large $n$, this can be approximated by
$$\frac{x^2}2=\log n.$$
Below, a plot of $\dfrac{x^2}2+\log x$ vs. $\dfrac{x^2}2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2606839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Spectral decomposition of some special matrix Let $A_{n\times n} = aI+bJ$, where $I$ is the identity matrix and $J$ is the matrix of all ones. Is it possible to find the expression of $A^{1/2}$ such that $A^{1/2}A^{1/2} = A$? In particular $A = I_{n\times n} - \frac{(1-\alpha)}{n+\alpha(2-n)}J_{n\times n}$, where $0<\a... | Here is a quick and easy way to get a matrix $A^{1/2}$, provided one is willing to limit oneself to solutions of the form
$C = \alpha I + \beta J; \tag 1$
then
$C^2 = \alpha^2 I + 2 \alpha \beta J + \beta^2 J^2 = \alpha^2 I + 2 \alpha \beta J + \beta^2 n J = \alpha^2 I + (2 \alpha \beta + n \beta^2)J, \tag 2$
where we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
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.