Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
How to find $\int \frac{e^{-x^2}}{x^2 + 1} dx$? I have a question about improper integrals:
How can we find $\lim_{n \rightarrow +\infty}\int_{-n}^{n} \frac{1 - e^{-nx^2}}{x^2(1+nx^2)}dx$?
$\textbf{Some effort:}$
$\lim_{n \rightarrow +\infty}\int_{-n}^{n} \frac{1}{n} \frac{1 - e^{-nx^2}}{x^2(1+nx^2)}dx = \lim_{n \right... | $$I_n=\int_{-n}^{n} \frac{1 - e^{-nx^2}}{x^2(1+nx^2)}\,dx=2\int_{0}^{n} \frac{1 - e^{-nx^2}}{x^2(1+nx^2)}\,dx$$ Let $$nx^2=t\implies x=\frac{\sqrt{t}}{\sqrt{n}}\implies dx=\frac{dt}{2 \sqrt{n} \sqrt{t}}$$ making
$$I_n=\sqrt{n}\int_0^1\frac{ \left(1-e^{-t}\right)}{ t^{3/2} (t+1)}\,dt$$ You do not need to compute anythi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2556615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
If $x,y,z\in {\mathbb R}$, Solve this system equation: If $x,y,z\in {\mathbb R}$, Solve this system equation:
$$
\left\lbrace\begin{array}{ccccccl}
x^4 & + & y^2 & + & 4 & = & 5yz
\\[1mm]
y^{4} & + & z^{2} & + & 4 & = &5zx
\\[1mm]
z^{4} & + & x^{2} & + & 4 & = & 5xy
\end{array}\right.
$$
This is an olympi... | Your system can be written as follows
$$
\left\lbrace\begin{array}{clc}\tag{1}
\left( {x}^{2}-2 \right) ^{2}+ \left( y-z \right) ^{2}+ \left( x-z
\right) \left( x+z \right) +3\,({x}^{2}-\,zy)&=&0
\\ \\[1mm]
\left( {y}^{2}-2 \right) ^{2}+ \left( z-x \right) ^{2}+ \left( y-x
\right) \left( y+x \right) +3(\,{y}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2556960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Evaluate $\int_0^\infty \left(\frac{x}{\sinh x}\right)^3dx$ I need to evaluate $$\int_0^\infty \left(\frac{x}{\sinh x}\right)^3dx$$
I know that I need to use the residue theorem to solve it, but I don't understand how to choose contour properly.
Thank you for any help!
| Integrate by parts
\begin{align}
I& = \int_0^\infty \left(\frac{x}{\sinh x}\right)^3dx\\
&= -\int_0^\infty x^3 \text{csch} \>x \>d(\coth x)\\
&= -3\int_0^\infty x^2 d(\text{csch} \>x)
-\int_0^\infty x^3 \text{csch} \>x \coth^2 xdx\\
&= 6\int_0^\infty x \>\text{csch} \>x \>dx
-\int_0^\infty x^3 \text{csch} \>x \>dx -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2562153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Solution of the ordinary differential equation $y'(t)=-y^3+y^2+2y$ Consider the solution of the ordinary differential equation $y'(t)=-y^3+y^2+2y$ subject to $y(0)=y_0 \in (0,2). $ then $\lim \limits_{t \to \infty}y(t)$ belongs to
*
*{-1,0}
*{-1,2}
*{0,2}
*{0, $+\infty $}
My Attempt: $y'(t)=-y^3+y^2+2y \\ \Rig... | you got $\frac{(y-2)(y+1)^2}{y^3}=\frac{(y_0-2)(y_0+1)^2}{y_0^3} e^{-6t}$ take the limit and you get $$\lim\limits_{t\to\infty}\frac{(y-2)(y+1)^2}{y^3}=0\implies \lim\limits_{t\to\infty}(y-2)(y+1)^2=0 \implies \lim\limits_{t\to\infty}y(t)=\begin{cases}2\\-1\end{cases}$$ hence option (2) is true
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2563252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solving: $(x-a)(x-b)=x-c$, $(x-c)(x-b)=x-a$ and $(x-c)(x-a)=x-b$ Given three distinct real numbers $a$, $b$, and $c$, show that at least two of the three following equations
$$(x-a)(x-b)=x-c$$
$$(x-c)(x-b)=x-a$$
$$(x-c)(x-a)=x-b$$
have real solutions.
My attempt: I tried to multiply all the equations side by side to ob... | Let the three quadratics be
$$\begin{align}
p_1(x)&=(x-a)(x-b)-(x-c)\\
p_2(x)&=(x-b)(x-c)-(x-a)\\
p_3(x)&=(x-c)(x-a)-(x-b)\\
\end{align}$$
By symmetry, we may assume $a\le b\le c$. But that implies
$$p_2(b)=-(b-a)\le0\quad\text{and}\quad p_3(c)=-(c-b)\le0$$
Thus $p_2$ and $p_3$ have at least one real root (since their... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2563590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove the inequality $\frac{b+c}{a(y+z)}+\frac{c+a}{b(z+x)}+\frac{a+b}{c(x+y)}\geq \frac{3(a+b+c)}{ax+by+cz}$
Suppose that $a,b,c,x,y,z$ are all positive real numbers. Show that
$$\frac{b+c}{a(y+z)}+\frac{c+a}{b(z+x)}+\frac{a+b}{c(x+y)}\geq \frac{3(a+b+c)}{ax+by+cz}$$
Below are what I've done, which may be misleadi... | This is not an answer, but putting code in a comment doesn't work well either.
Regarding your doubts about the truth of the inequality, I am still inclined to think it holds true. You can try to look for counterexamples with this code. In case python thinks the inequality doesn't hold, it outputs $a,b,c,x,y,z$ and the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 2
} |
Quadratic equation with different indices My maths teacher gave me a worksheet to work through as I was getting slightly bored in lessons. However, there was one question which I cannot do. The worksheet gives the answer, but you are supposed to show how you did it. Here is the question:
$729 + 3^{2x+1} = 4\times3^{x... | HINT
Set $3^x=y$
$$729 + 3y^2= 4\cdot 9 y$$
$$3y^2-36y+729=0$$
$$y^2-12y+243=0$$
$\Delta <0 \implies$ no real solution
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Integer solutions of $2x+3y=n$ What is the smallest $m$ such that for all $n\geq m$, the equation $2x+3y=n$ has solutions with $x,y \in \mathbb{Z}$ and $x,y\geq2$?
My approach. We can write the solutions in terms of the parameter $t$ as:
$$x(t)=-n-3t$$ and
$$y(t) = n+2t$$
Setting both of those greater than or equal to... | You are almost there.
Solving for $t$, we find $\dfrac{-n-2}{3} \geq t \geq\dfrac{2-n}{2}$. Solving the resulting inequality, we get
$$-2n-4\geq6-3n$$
So $n\geq10$.
You want to find $n$ such that there is at least one integer $t$ satisfying
$$\dfrac{-n-2}{3} \geq t \geq\dfrac{2-n}{2}\tag1$$
For this, solving $\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How to prove $\{n!e\} = \frac{1}{n +1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \dots$ I have the definition $a_n = \frac{1}{n +1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \dots$
I need to show that:
$a)$ $0 < a_n < \frac{1}{n}$
$b)$ $a_n = n!e - \lfloor{n!e}\rfloor$
So I know that
$\frac{1}{n} ... | $$n!e=n!\sum_{k=0}^\infty\frac1{k!}=\sum_{k=0}^\infty\frac{n!}{k!}$$
Since $k!$ divides $n!$ for $k\le n$, the sum
$$\sum_{k=0}^n\frac{n!}{k!}$$
is an integer, say $N$. Then
$$n!e=N+\sum_{k=n+1}^\infty\frac{n!}{k!}=\sum_{k=1}^\infty\frac1{\prod_{j=1}^k(n+k)}<\sum_{k=1}^\infty\frac1{(n+1)^k}=\frac1n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Is this a valid method of solving $3^x + 2^x = 35$? I want to know if this is a valid method of solving this equation:
$3^x + 2^x = 35$
$3^x+2^x = (7)(5)$
$3^x+2^x = (3+2^2)(2+3)$
$3^x+2^x = 3^2 + 2^3 + 18$
$3^x+2^x = 9 + 18 + 2^3 $
$3^x+2^x = 27 + 2^3 $
$3^x+2^x = 3^3 + 2^3 $
And now comes my problem. Is it correct to... | The claim is equivalent to
$$
f(x)=f(3)\implies x=3
$$
where $f(x)=3^x+2^x$. This is true because $f$ is injective (since it is strictly increasing for example).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
How and where can I calculate constant $k\approx1,895$? We know, that
$$e=\lim\limits_{n\to\infty}^{}\left[1+\frac{1}{1}\left(1+\frac{1}{2}\left(1+\frac{1}{3}\left(1+\cdots\frac{1}{n}\right)\right)\cdots\right)\right]$$
If $x_{m}=m(x_{m-1}-1)$ and $x_{0}=e$, then
$$x_{1}=\lim\limits_{n\to\infty}^{}\left[1+\frac{1}{2}\l... | For general $m \in \mathbb{N}$, it is easy to see
$$x_m = m!\left(e - \sum_{k=0}^{m-1}\frac{1}{k!}\right)
\quad\implies\quad x_m - 1
= m!\left(e - \sum_{k=0}^{m} \frac{1}{k!}\right)
= \sum_{k=m+1}^\infty \frac{m!}{k!}$$
This leads to
$$\begin{align}x_m - 1
= \sum_{k=0}^\infty \frac{m!}{(m+k+1)!}
&= \frac{1}{m+1} + \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
using a base 3 decimal to express as a base 10 fraction using geometric series
Express $0.\overline{21}_3$ as a base 10 fraction in reduced form.
So I was able to solve it by setting $x=\overline{.21}$, but the solution also briefly mentioned another way using the geometric series:
A quick way to get the answer by u... | \begin{align}
(0.212121 \ldots)_3
&= \dfrac{21_3}{100_3}
+ \dfrac{21_3}{(100_3)^2}
+ \dfrac{21_3}{(100_3)^3} + \cdots\\
&= \frac 79 + \frac{7}{9^2} + \frac{7}{9^3} + \cdots \\
&= \frac 79\left( \dfrac{1}{1-\frac 19} \right) \\
&= \frac 79 \cdot \frac 98 \\
&= \frac 78.
\end{align}
You coul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Does $\sum\limits_{k=2}^\infty\frac{(-1)^{2^k}x^{2k}}{(2k+3)!}$ converge absolutely? Let $\sum\limits_{k=2}^\infty\frac{(-1)^{2^k}x^{2k}}{(2k+3)!}$, $x\in\mathbb{R}$. Does this series converge absolutely?
$$\sum\limits_{k=2}^\infty\frac{(-1)^{2^k}x^{2k}}{(2k+3)!}$$ converges absolutely if $$\sum\limits_{k=2}^\infty\lef... | Yes since $$\sinh x = \sum\limits_{k=
0}^\infty\frac{x^{2k+1}}{(2k+1)!}$$
then $$ \sum\limits_{k=
2}^\infty\frac{x^{2k}}{(2k+3)!}\le \sum\limits_{k=
2}^\infty\frac{x^{2k}}{(2k+1)!}= -1-\frac{x^2}{6}+ \sum\limits_{k=
0}^\infty\frac{x^{2k}}{(2k+1)!}\\=-1- \frac{x^2}{6}+ \frac{1}{|x|}\sum\limits_{k=
0}^\infty\frac{|x|^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Evaluate $\int \frac{a^2\sin^2 x+b^2\cos^2 x}{a^4\sin^2 x+b^4\cos^2 x}dx$ Evaluate
$$\int \frac{a^2\sin^2 x+b^2\cos^2 x}{a^4\sin^2 x+b^4\cos^2 x}\text dx$$
I would have given my attempt to this question but honestly, I think my attempts to solve this did nothing but only complicated it further.
Any hints or suggestions... | Write $$\begin{align}\frac{a^2\sin^2 x+b^2\cos^2 x}{a^4\sin^2 x+b^4\cos^2 x}&=\frac1{a^2+b^2}\cdot \frac{(a^4+a^2b^2)\sin^2 x+(b^4+a^2b^2)\cos^2 x}{a^4\sin^2 x+b^4\cos^2 x}\\&=\frac1{a^2+b^2}\cdot \frac{a^4\sin^2 x+b^4\cos^2 x+a^2b^2}{a^4\sin^2 x+b^4\cos^2 x}\\&=\frac1{a^2+b^2}\cdot \left(1+\frac{a^2b^2}{a^4\sin^2 x+b^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Show that $f(x,y,z)=6x^2+4y^2+2z^2+4xz-4yz \geq 0$ for all $x, y, z$ except for $x=y=z=0$ When I try to factor the quadratic form, I end up with
$$6x^2+4y^2+2z^2+4xz-4yz = 2((x+z)^2+2x^2+(y-z)^2-z^2)$$
which does not ensure that $f(x,y,z) \geq 0$ for all $x, y, z \geq 0$ since the $z$ term is negative. How should the... | Your quadratic form is positive definite. I do not know what the eigenvalues of the matrix are. This method is the same as "completing the square." The diagonal entries of the diagonal matrix $D$ are all positive.
$$ P^T H P = D $$
$$\left(
\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
- \frac{ 1 }{ 3 } & \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
} |
$a+b \sqrt{2}=(1+ \sqrt{2})^n$ when $a^2-2b^2=+1 or -1$ I want to prove this statement :
$a+b \sqrt{2}=(1+ \sqrt{2})^n$ when $a^2-2b^2=+1 or -1$
Where a,b,and n are positive integers .
Usually people consider it true saying $1+ \sqrt{2}$ is fundamental unit of $\mathbb{Z} [ \sqrt{2}]$ , but how can I proof this more ... | Define $a_n$ and $b_n$ by
\begin{eqnarray*}
a_n+b_n \sqrt{2} =(1+\sqrt{2})^n.
\end{eqnarray*}
Their recurrence relation is easily derived by
\begin{eqnarray*}
a_{n+1}+b_{n+1} \sqrt{2} =(a_n+b_n \sqrt{2}) (1+\sqrt{2}) \\
a_{n+1}=a_n +2 b_n \\
b_{n+1} =a_n+b_n
\end{eqnarray*}
$(a_n,b_n)$ can be shown to satisfy Pell's ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Multiplying out $(x^3 + 3x^2 + 3x + 1) \times (1 + x + x^2)$ I know this is a basic math question but still I always get stuck on this.
$(x^3 + 3x^2 + 3x + 1) \times (1 + x + x^2)$
I know this is the solution:
$1 + 4x + 7x^2 + 7x^3 + 4x^4 + x^5$
But I want to know how they got there.
This has to be solved with distrib... | Yes, you are right, so far. But, why can’t you just sum up terms with same power of $x$, that is, $$(x^3+3x^2+3x+1)(1+x+x^2) $$ $$= (x^\color{red}{5})+(3x^\color{green}{4} + x^\color{green}{4}) + (x^\color{blue}{3}+3x^\color{blue}{3}+3x^\color{blue}{3})+(3x^\color{orange}{2}+3x^\color{orange}{2}+x^\color{orange}{2})+(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why does multiplying by $\textbf{A}^T$ make a previously unsolvable linear system solvable Consider for instance the linear system:
$$\left(
\begin{array}{cc}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{array}
\right).\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)=\left(
\begin{array}{c}
1 \\
2 \\
4 \\
\end{array}
\... |
I assumed that the solutions to $\textbf{A}x = \textbf{B}$ were the same as the solutions to $\textbf{P}\textbf{A}x = \textbf{P}\textbf{B}$.
The direct implication $\textbf{A}x = \textbf{B} \implies \textbf{P}\textbf{A}x = \textbf{P}\textbf{B}$ holds for any $\textbf{P}$. However, the reverse implication $\textbf{P}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2583454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Find: $ \lim_{x\to \infty}\frac{x-1}{\sqrt{x}}-\frac{x-1}{\sqrt{x+1}}$
Find: $\displaystyle \lim_{x\to \infty} f(x)=\frac{x-1}{\sqrt{x}}-\frac{x-1}{\sqrt{x+1}}$
The answer provided in the book is 0 (also checked in Wolfram Alpha), but I can't find a good argument (without L'Hopital), to prove that. I end up in a $\in... | $$\frac{x-1}{\sqrt{x}}-\frac{x-1}{\sqrt{x+1}}=\frac{(x-1)\sqrt{x+1}-(x-1)\sqrt{x}}{\sqrt{x}\sqrt{x+1}}=(x-1)\frac{\sqrt{x+1}-\sqrt{x}}{\sqrt{x}\sqrt{x+1}}\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}=\frac{x-1}{\sqrt{x}\sqrt{x+1}(\sqrt{x+1}+\sqrt{x})}\to0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 3
} |
Generalized Formula for the Probability of the Union of $n$ events occurring? Consider
$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$
What is the generalization of this formula for $n$ events occuring? That is
$$
P(\cup A_i) = \sum P(A_i) + \ldots ?
$$
| We have
\begin{align}
P(A_1\cup A_2\cup A_3)= & \;\;\;\;P(A_1)+P(A_2) +P(A_3)
\\
&
-\underbrace{P(A_1\cap A_{2})}_{1<2}
-\underbrace{P(A_1\cap A_{3})}_{1<3}
-\underbrace{P(A_2\cap A_{3})}_{2<3}
\\
&
+\underbrace{P(A_1\cap A_{2}\cap A_{3})}_{1<2<3}
\end{align}
and
\begin{align}
P(A_1\cup A_2\cup A_3\cup A_4)= & \;\;\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2587979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find the range of $f(x)=11\cos^2x+3\sin^2x+6\sin x\cos x+5$ I'm trying to solve this problem.
Find the range of $f(x)=11\cos^2x+3\sin^2x+6\sin x\cos x+5$
I have simplified this problem to $$f(x)= 8\cos^2x+6\sin x\cos x+8$$ and tried working with $g(x)= 8\cos^2x+6\sin x\cos x$.
I factored out the $2\cos x$ and rewrote... | Hint:
Let $y=A\sin^2x+B\cos^2x+C\sin x\cos x+D$
Method$\#1:$ Divide both sides by $\cos^2x$ to form a Quadratic Equation in $\tan x$
As $\tan x$ is real, the discriminant must be $\ge0$
Method$\#2:$ Divide both sides by $\sin^2x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Prove that $\sum^m_{k=0}\binom{m}{k}(-1)^k(\frac{1}{n+k+1}) = \frac{m!n!}{(m+n+1)!}$ I am looking for a more direct proof of the following identity: $$\sum^m_{k=0}\frac{\binom{m}{k}(-1)^k}{n+k+1} = \frac{m!n!}{(m+n+1)!}$$
My original proof comes from evaluating $\int^1_0{x^n(1-x)^m}{dx}$, $n, m \in \mathbb{N}_0$ in two... | Here is a more direct approach along the lines you were perhaps thinking.
Notice that
$$\int_0^1 x^{n + k} \, dx = \frac{1}{n + k + 1}.$$
With this result your sum can be rewritten as
$$S = \sum_{k = 0}^m \binom{m}{k} (-1)^k \int_0^1 x^{n + k} \, dx = \int_0^1 x^n \left [\sum_{k = 0}^m \binom{m}{k} (-1)^k x^k \right ] ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Continuity And Derivative Of A Function
Let $
f(x)=
\begin{cases}
\sin^2x \sin(\frac{1}{x}), x\neq 0\\
0, x=0\\
\end{cases}
$
a. Find all the points where $f(x)$ is continuous
b. Find $f'(x)$ for all the points $f(x)$ is continuous
c. Is $f'(x)$ continuous where it is defined? if not where and which discontinuity typ... | First, the product of continuously differentiable maps is continuously diufferentiable. As $x \mapsto \sin^2 x$ and $x \mapsto \sin \left( \frac{1}{x} \right)$ are both continuously differentiable for $x \neq 0$, $f$ is continuously differentiable for all $x \neq 0$.
Also, for all $x \in \mathbb R$, you have $0 \le \ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2602067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Integrate $\int \frac {x^4}{\sqrt {x^2-9}} \,dx$ Integrate $\int \dfrac {x^4}{\sqrt {x^2-9}} dx$
My Attempt:
Let $x=3\sec (\theta )$
$$dx=3\sec (\theta).\tan (\theta).d\theta$$
Then,
$$=\int \dfrac {x^4}{\sqrt {x^2-9}}$$
$$=\int \dfrac {81. \sec^4 (\theta)}{\sqrt {(3\sec (\theta))^2 - 9}} 3\sec (\theta).\tan (\theta).d... | To integrate $\int \sec^5 (\theta) d\theta$, use integration by parts.
u = $\sec^3 (\theta)$
dv = $\sec^2 (\theta) d\theta$
And so du = $\sec^3\theta\tan\theta d \theta$
and $v = \tan\theta$
Thus $$\int \sec^5 (\theta) d\theta = \sec^3 (\theta) \tan \theta - \int 3\sec^3\theta(sec^2\theta-1)d\theta = \sec^3 (\theta) \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2605920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove that $\frac{1}{\sqrt{ab+a+2}}+ \frac{1}{\sqrt{bc+b+2}}+ \frac{1}{\sqrt{ac+c+2}} \leq \frac{3}{2}$ Let $abc=1$ and $a,b,c>0$.
Prove that
$$\frac {1}{\sqrt {ab+a+2}}+ \frac {1}{\sqrt {bc+b+2}}+ \frac {1}{\sqrt {ac+c+2}} \leq \frac {3}{2}.$$
I guess that $$\frac {1}{\sqrt {ab+a+2}}+ \frac {1}{\sqrt {bc+b+2}}+ \frac... | Let $a=\frac{y}{x}$ and $b=\frac{z}{y}$, where $x$, $y$ and $z$ are positives.
Thus, $c=\frac{x}{z}$ and by C-S we obtain:
$$\sum_{cyc}\frac{1}{\sqrt{ab+a+2}}=\sum_{cyc}\sqrt{\frac{1}{\frac{z}{x}+\frac{y}{x}+2}}=\sum_{cyc}\sqrt{\frac{x}{2x+y+z}}$$
$$\leq\sqrt{\sum_{cyc}1^2\sum_{cyc}\frac{x}{2x+y+z}}=\sqrt{3\sum_{cyc}x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How can I prove that $7+7^2+7^3+...+7^{4n} = 100*a$ (while a is natural number)? How can I prove that $7+7^2+7^3+...+7^{4n} = 100*a$ (while a is entire number) ?
I thought to calculate $S_{4n}$ according to:
$$ S_{4n} = \frac{7(7^{4n}-1)}{7-1} = \frac{7(7^{4n}-1)}{6} $$
But know, I don't know how to continue for ge... | Induction proof of
$$\sum_{i=1}^{4n}7^i=100a.$$
For $n=1$, it is true:
$$7^1+7^2+7^3+7^4=2800=100\cdot 28.$$
Assuming it is true for $n$:, prove for $n+1$:
$$\begin{align}\sum_{i=1}^{4(n+1)}7^i=&\sum_{i=1}^{4n}7^i+7^{4n+1}+7^{4n+2}+7^{4n+3}+7^{4n+4} =\\ &100a+7^{4n+1}+7^{4n+2}+7^{4n+3}+7^{4n+4}
=\\ &100a+7^{4n}(7+7^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Show the inequality involving the product of distinct primes that divide $n$. Let $\displaystyle \omega (n)=\sum_{p\mid n} 1$ be the number of distinct primes dividing $n$. Show that
$\displaystyle \phi (n) \geq n \prod_{k=2}^{\omega (n) +1} (1 - \frac{1}{k}) = \frac{n}{\omega (n) +1}$ and that
$\displaystyle \phi (n)... | First, suppose that $n = \prod\limits_{k = 1}^{ω(n)} p_k^{a_k}$, then\begin{align*}
\frac{φ(n)}{n} &= \prod_{k = 1}^{ω(n)} \left( 1 - \frac{1}{p_k} \right) \geqslant \prod_{k = 1}^{ω(n)} \left( 1 - \frac{1}{k + 1} \right) = \prod_{k = 2}^{ω(n) + 1} \left( 1 - \frac{1}{k} \right)\\
&= \prod_{k = 2}^{ω(n) + 1} \frac{k - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\lim\limits_{x\to \infty}\frac{x^2}{2^x-1}$
I would like to evaluate
$$\lim_{x\to \infty}\frac{x^2}{2^x-1}$$
Without using L'HOSPITAL's rule nor series.
I tried more than one
technique such the sub
$$x=\frac{1}{y}$$
But i could not get the solution ?
| Using the hint provided by Jyrki Lahtonen in the comments, we may sneakily apply the binomial theorem in order to obtain
$$ 2^n - 1 = (1+1)^n - 1 = -1 + \sum_{k=0}^{n} \binom{n}{k} \ge \binom{n}{k} $$
for any choice of natural number $k$ less than $n$. This implies that
$$ \frac{n^2}{2^n - 1} \le \frac{n^2}{\binom{n}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Does $x+\sqrt{x}$ ever round to a perfect square, given $x\in \mathbb{N}$? I'll define rounding as $$R(x)=\begin{cases} \lfloor x \rfloor, & x-\lfloor x \rfloor <0.5 \\ \lceil x \rceil, & else\end{cases}$$
Does $x+\sqrt{x}$ ever round (to the nearest integer) to a perfect square, given $x\in \mathbb{N}$?
For example... | Since
$$
\overbrace{n^2-n+\sqrt{n^2-n}}^{\text{$n^2-n$ is too small}}\lt n^2-\frac12\iff\overbrace{\sqrt{n^2-n}\lt n-\frac12}^{n^2-n\,\lt\,n^2-n+\frac14}
$$
and
$$
\overbrace{n^2-n+1+\sqrt{n^2-n+1}}^{\text{$n^2-n+1$ is too big}}\gt n^2+\frac12\iff\overbrace{\sqrt{n^2-n+1}\gt n-\frac12}^{n^2-n+1\,\gt\,n^2-n+\frac14}
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
Evaluating: $\int \frac {1+\sin (x)}{1+\cos (x)} dx$
Evaluate: $\int \dfrac {1+\sin (x)}{1+\cos (x)} dx$
My Attempt:
$$=\int \dfrac {1+\sin (x)}{1+\cos (x)} dx$$
$$=\int \dfrac {(\sin (\dfrac {x}{2}) + \cos (\dfrac {x}{2}))^2}{2\cos^2 (\dfrac {x}{2})} dx$$
$$=\dfrac {1}{2} \int (\dfrac {\sin (\dfrac {x}{2}) + \cos (\... | Here is another trick
$$
\begin{align}
I&=\int \dfrac {1+\sin (x)}{1+\cos (x)} dx\\
&=\int \dfrac {1+ \sin(0)+\sin (x)}{\cos(0)+\cos (x)} dx\\
&=\int \dfrac {1+ 2\sin(x/2)\cos (x/2)}{2\cos^2(x/2)}dx\\
&=\int \dfrac {1}{2\cos^2(x/2)}dx+\int \dfrac {\sin(x/2)}{\cos(x/2)}dx\\
&=\int \dfrac {1}{2\cos^2(x/2)}dx+\int {\tan(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 4
} |
Find $\sum_{n=1}^{\infty} n\left(2n-1\right)x^{2n}$
Question: Find $$\sum_{n=1}^{\infty} n\left(2n-1\right)x^{2n}.$$
My Approach:$$
\lim_{n\rightarrow\infty} |n\left(2n-1\right)|^{\frac{1}{n}} =1 \Longrightarrow \text{Radius
of convergence} = 1.
$$
I don't know how to find the sum. Any Hint or help will be highly a... | Take $f(x)=\sum_{n=1}^{\infty}n(2n-1)x^{2n}$ therefore:$$g(x)=\int \frac{f(x)}{x^2}dx=C_1+\sum_{n=1}^{\infty}nx^{2n-1}\\g_1(x)=\sum_{n=1}^{\infty}nx^{2n-1}\\h(x)=\int \frac{g_1(x)}{x}dx^2=C_2+\sum_{n=1}^{\infty}x^{2n}=\frac{1}{1-x^2}-1+C_2$$Now by successive differentiation we obtain:$$g_1(x)=x\frac{dh(x)}{d(x^2)}=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2619140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is this true: $\lim_{x\to 0} \frac{1-(\cos x)^n}{x^2}=\frac n2$ Does this statement hold $\forall n\in \Bbb R$?
$$\lim_{x\to 0} \frac{1-(\cos x)^n}{x^2}=\frac n2$$
I know that it holds for $\forall n \in \Bbb N$, because $1-(\cos x)^n=(1-\cos x)(1+ \cos x + ... + (\cos x)^{n-1})$, and $\lim_{x\to 0} (1+ \cos x + ... + ... | Note that for $n\in \mathbb{N}$
$$1-(\cos x)^n=(1-\cos x)\stackrel{\color{red}{\text{n terms}}}{(\cos^{n-1}x+\cos^{n-2}x+...+1)}$$
thus since $\cos x\to 1$
$$\frac{1-(\cos x)^n}{x^2}=\frac{(1-\cos x)}{x^2}\stackrel{\color{red}{\text{sum up to n}}}{(\cos^{n-1}x+\cos^{n-2}x+...+1)}\to\frac12\cdot n=\frac n2$$
For $n=a\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2625644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Solving a biquadratic.
If $x$ is a real number and satisfies $$x+ \sqrt[4] {5-x^4}=2$$ then find the value of $$x\sqrt[4] {5-x^4}$$
My try :
The question is significantly asking for the value of $-x(x-2)$ if we get the root of the equation $$2x^4-8x^3+24x^2-32x+11=0$$
Using this I have reached till
$$-2x(x-2)(x^2-2... | Note that
$$2x^4-8x^3+24x^2-32x+11=0 \implies 2 (x^2 - 2 x)^2 + 16 (x^2 - 2 x) + 11=0$$
now take $x^2 - 2 x=y$ and solve
$$2 y^2 + 16y+ 11=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627661",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 1
} |
Finding a matrix with conditions on $A^{-1}$ and $\det(A)=0.5$
Find matrix $A$ (without guessing) if $A^{-1}$ $2^{\mkern1mu\text{nd}}$ column is\begin{pmatrix} 2 \\ 1 \\-1 \end{pmatrix}
And $\det(A) = 0.5$
Attempt -
I could make a progress knowing that $\det(A^{-1}) = 2$ obviously.
I could also notify that if $A = $... | I would proceed like this $A^{-1}=\begin{pmatrix} a & 2 & d\\ b & 1 & e \\ c & -1 & f\end{pmatrix}$
The only additional constraint we have is $\det(A^{-1})=af+ae-bd-2bf+2ce-cd=2$
Since we are required to find an instance of $A$ that works and not all possible matrices, then just select some values of $(a,b,c,d,e,f)$ th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Beckenbach Introduction to Inequalities Chapter 2: Show $(a+b)/2 \le ( (a^2 + b^2 )/2)^{1/2}$ I'm having trouble understanding the following problem
Problem
Beckenbach, Chapter 2 Pg 24 Ex 1
$$
\text{Show the following for all a, b}\quad
\frac{(a+b)}{2} \le \left(\frac{a^2 + b^2}{2}\right)^\frac{1}{2}
$$
The book pr... | Squaring both sides...
$\cfrac{(a+b)^2}{4}\le \cfrac{a^2+b^2}{2}$
$\cfrac{a^2+2ab+b^2}{4}\le \cfrac{a^2+b^2}{2}$
$2a^2+4ab+2b^2\le4(a^2+b^2)$
$2a^2+4ab+2b^2\le4[(a+b)^2-2ab]$
$(a+b)^2\le2(a+b)^2-4ab$
Which proves your inequality, since any numbers $0<x<1$ will produce a positive result, as $-4ab$ will equal to a value ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Checking the differentiability of the following function
Check the differentiability of the following function $$f(x)=(x+1)|x^2-1|$$ at points $x=1$ and $x=-1$.
My approach
I have written the function in the following form:
$$f(x)=\begin{cases}
x^3-x+x^2-1 & \text{ if } x\leq-1,x\geq1 \\
x-x^3+1-x^2 & \text{ if } -... | Not differentiable at $x= 1:$
$f(x)=(x+1)|(x+1)(x-1)| = (x+1)^2 |x-1|$
for $x>0.$
Consider : $\dfrac{f(x)-f(1)}{x-1} =$
$\dfrac{(x+1)^2|x-1| }{x-1}.$
$\lim_{x \rightarrow 1^-} \dfrac{(x+1)^2(1-x)}{x-1}=-4.$
$\lim_{x \rightarrow 1^+} \dfrac{(x+1)^2(x-1)}{x-1}= 4.$
Not differentiable at $x=1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
How to prove $\tan\Big[\frac{1}{2}\sin^{-1}\frac{3}{4}\Big]=\frac{4-\sqrt{7}}{3}$ Prove
$$
\tan\Big[\frac{1}{2}\sin^{-1}\frac{3}{4}\Big]=\frac{4-\sqrt{7}}{3}
$$
and justify why $\frac{4+\sqrt{7}}{3}$ is ignored.
My Attempt:
$$
\tan x=\frac{2\tan\frac{x}{2}}{1-\tan^2\frac{x}{2}}\implies\tan x-\tan x\tan^2\frac{x}{2}... | let $\sin y= \frac{3}{4}$ we need to prove
$\tan \left[\frac{1}{2}\sin^{-1} \left(\frac{3}{4}\right)\:\right]=\frac{4-\sqrt{7}}{3}$
otherwise seen as
$\tan \left[\frac{1}{2}y\:\right]=\frac{4-\sqrt{7}}{3}$
We also know the identity
$\tan \left[\frac{1}{2}y\:\right]=\frac{1-\cos\left[y\:\right]}{\sin \left[y\:\right]}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2632633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Solve $f'_x-xf'_y=y.$
Find the solution to $$\frac{\partial}{\partial x}f(x,y)
-x\frac{\partial}{\partial y}f(x,y) = y$$
that satisfies $f(x,0)=x^2+\frac{x^3}{3},$ by using the transformation
$$\left\{ \begin{array}{rcr}
u & = & ax^2+y \\
v & = & x \\ \end{array} \right.$$
for an appropriate value of ... | $$\begin{align}
f(x,y)= & \left(\frac{x^2}{2}+y\right)x-\frac{x^3}{6}+g\left(\frac{x^2}{2}+y\right)\\
f(x,0)= &\left(\frac{x^2}{2}+0\right)x-\frac{x^3}{6}+g\left(\frac{x^2}{2}+0\right) \\
=& \frac{x^3}{2}-\frac{x^3}{6}+g\left(\frac{x^2}{2}\right) = \frac{x^3}{3}+g\left(\frac{x^2}{2}\right) \neq \frac{x^2}{2}-\frac{x^3}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2636123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove the positivity of a sequence Let $a>0$ a real number and $(u_n)$ the sequence defined by
$$
u_{n+1} = a - \frac{1}{u_n}\text{ and } u_0 = a.
$$
Question: Determine condition on the value of $a>0$ such that the sequence $(u_n)$ is always positive.
Attempt: I tried to establish a general formula of $u_n$ in order... | Hint: $\;\require{cancel}
u_{n+1} - u_n = \left(\cancel{a} - \dfrac{1}{u_n}\right) - \left(\cancel{a} - \dfrac{1}{u_{n-1}}\right)=\dfrac{u_n-u_{n-1}}{u_nu_{n-1}}
\,$, so as long as the terms are positive the differences between consecutive terms have the same sign i.e. the sequence is monotonic, and is in fact decreas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Probability of Maximum Value
Let $X$ and $Y$ be independent, each uniformly distributed on $\{1, 2, ..., n\}$. Find $P(\max(X, Y) = k)$ for $1 \le k \le n$.
$P(\max(X, Y) = k) $
$= P(X = k \cap X > Y) + P(Y = k \cap Y > X)$
$= P(X=k)P(X>Y|X=k) + P(Y=k)P(Y>X|Y=k)$
$= \frac{1}{n} \cdot \frac{k-1}{n} + \frac{1}{n} \cdot... | $P(\max(X, Y) = k)$
$= P(X = k \cap X > Y) + P(Y = k \cap Y > X) + P(X = Y = k)$
$= P(X=k)P(X>Y|X=k) + P(Y=k)P(Y>X|Y=k) + P(X = Y = k)$
$= \frac{1}{n} \cdot \frac{k-1}{n} + \frac{1}{n} \cdot \frac{k-1}{n} + \frac1{n^2}$
$= \frac{2k-1}{n^2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2639134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve $yy' + x =\sqrt{x^2 + y^2}$ - Substitution for Diff Eqs. I have this problem
$$yy' + x = \sqrt{x^{2} + y^{2}}$$
I wanted to know if my work and answer are correct.
Let $v = x^{2} + y^{2}. v' = 2x + 2yy'$
$$ \frac{v'}{2} = x + yy'$$
Substituting everything in gives me:
$$\frac{v'}{2} = \sqrt{v}$$
From here it beco... | Your solution is correct.
At $$y^2 = \bigg(x+\frac{C}{2}\bigg)^{2} - x^2$$
You may simplify $$ y^2 = \bigg(x+\frac{C}{2}\bigg)^{2} - x^2=Cx+\frac {C^2}{4}$$
Which is a family of parabolas.$$y^2 =Cx+\frac {C^2}{4}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2639510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How show that $40 \mid n.$ Let $n\in\mathbb N$ s.t. $2n+1$, $3n+1$ are squares. Show that $40 \mid n.$
I have shown that $8 \mid n.$ Please help me to show $5 \mid n.$
| For $5 | n$ we have:
$3n+1=a^2 ⇒ 3n=(a-1)(a+1)$
$a-1$ and $a+1$ are two consecutive numbers, therefore n must also be consecutive to 3 that is $n=5$ or $5 | n$
Also:
$2n+1 =b^2 ⇒ 2n=(b-1)(b+1)$
$b-1$ and $b+1$ are consecutive numbers therefore n must also be consecutive to 2 that is $n=4$ or $4|n$ . So n can be the lea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why does $\sqrt[4]{-2}=\frac{1+i}{\sqrt[4]{2}}$? When I type $\sqrt[4]{-2}$ into WolframAlpha, it lists $\frac{1+i}{\sqrt[4]{2}}$ as an alternate form. How did it get that? I was trying to solve for $x^{4}+2=0$, and so I guess I could just say $\sqrt[4]{-2}$ is a solution, but I want it to look like $a+bi$ for $a,b\in\... | $$\left(\frac{1+i}{\sqrt[4]{2}}\right)^4=\frac{(1+i)^4}{2}=\frac{1}{2}\left(\sqrt{2}\left(\frac{1+i}{\sqrt{2}}\right)\right)^4=\frac{4}{2}\left(\frac{1+i}{\sqrt{2}}\right)^4=2\left(\exp\left(\frac{i\pi}{4}\right)\right)^4=$$
$$=2\exp(i\pi)=2*(-1)=-2$$
I've used these:
*
*$$\exp\left(\frac{i\pi}{4}\right)=\cos\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Sum of the series $\csc^{-1} \sqrt{10}+ \csc^{-1} \sqrt{50}+\csc^{-1}\sqrt{170}...$
Find the sum of the series:
$\csc^{-1}\sqrt{10}+ \csc^{-1}\sqrt{50}+\csc^{-1}\sqrt{170}...\csc^{-1}\sqrt{(n^2+1)(n^2+2n+2)}$
I converted the series to $\sum^{n} _{i=0}\arcsin \dfrac{1}{\sqrt{(i^2+1)(i^2+2i+2)}}$ and then tried to u... | The given sum can be represented as $$\tan ^{-1} \frac {1}{3}+\tan ^{-1} \frac {1}{7}+\tan ^{-1} \frac {1}{13}\cdots \tan ^{-1} \frac {1}{n^2+n+1}$$
This can be represented as $$\sum_{k=1}^n \tan ^{-1} \left(\frac {1}{k^2+k+1}\right)$$
$$\sum_{k=1}^n \tan ^{-1} \left(\frac {(k+1)-(k) }{1+(k+1)(k)}\right) $$
$$\sum_{k=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $A\in M_3\left(\mathbb C\right)$ is an invertible matrix such that $2A^2=4A+A^3$ If $A\in M_3\left(\mathbb C\right)$ is an invertible matrix such that
$$2A^2=4A+A^3$$
Then which of the following is(are)correct:
(A) $\det (A)=8;$
(B) $det\left(adj\left(\frac A2\right)\right)=1;$
(C) $tr\left(A-2I_3\right)^3=24;$
(D)... | From the given information, $A(A^2-2A+4I_3)=O_3$
$\implies A^2-2A+4I_3=O_3$
$\implies (A+2\omega I_3)(A+2\omega^2 I_3)=O_3$
$\implies$ eigen values of $A$ can be $-2\omega,-2\omega,-2\omega^2$ or $-2\omega,-2\omega^2,-2\omega^2.$ Where $\omega$ is a non real cube root of unity.
$\implies \det A=$ Product of eigen value... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2645051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove that $x^2yz+xy^2z+xyz^2 \leq \frac{1}{3}$ where $x^2+y^2+z^2 = 1$ and $x,y,z >0$? How to prove that $x^2yz+xy^2z+xyz^2 \leq \frac{1}{3}$ where $x^2+y^2+z^2 = 1$ and $x,y,z >0$?
| You can write the inequality as
$$
xyz(x+y+z)\leq\frac13.
$$
On the left-hand-side, using Cauchy-Schwarz,
$$
xyz(x+y+z)\leq xyz(x^2+y^2+z^2)^{1/2}(1+1+1)^{1/2}=\sqrt3\,xyz.
$$
Now we get the seemingly easier problem of maximizing $xyz$ under $x^2+y^2+z^2=1$. Here symmetry, common sense, or Lagrange multipliers show... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2648962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Minimum value of expression containing $2$ variables $x,y$
Finding minimum of $\displaystyle f(a,b)= (a-b)^2+\bigg(\frac{a^2}{20}-\sqrt{(17-b)(b-13)}\bigg)^2,$
where $a\in\mathbb{R^{+}},b\in(13,17)$
(Without using Geometry)
Try: Using partial derivative with r to $a$ (Treating $b$ as a constant)
$\displaystyle f... | As analysis teaches us, take the derivatives with respect to $a$ and $b$ separately, and converge them into a system of equations imposing them to be zero:
$$\begin{cases}
\large \frac{a^3}{100}+a \left(2-\frac{1}{5} \sqrt{-b^2+30 b-221}\right)-2 b = 0\\\\
\large \frac{a^2 (b-15)}{10 \sqrt{-b^2+30 b-221}}-2 a+30 = 0
\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2649341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solving $\arccos (x) +\arccos (2x)=\frac{3\pi}{4}$ $\arccos (x) +\arccos (2x)=\frac{3\pi}{4}$
$\implies \arccos (x)=\frac{3\pi}{4}-\arccos (2x)$
$\implies \cos(\arccos (x))=\cos(\frac{3\pi}{4}-\arccos (2x))$
$\implies x=\cos(\frac{3\pi}{4})\cos(\arccos (2x))+\sin(\frac{3\pi}{4})\sin(\arccos(2x))$
$\implies x=\cos(\frac... | $$\arccos(y)>\dfrac\pi2$$ for $y<0$
Here what if $x<0\iff2x<0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Finding the limit of the recursive sequence defined as $5a_{n+2}=3a_{n+1}+2a_n$
Find the limit of the recursive sequence defined as follows: $$5a_{n+2}=3a_{n+1}+2a_n\;\;\text{for $n\geq 0$, $a_1=1/2$ and $a_2=2/3$}.$$
Could someone give me a hint as to how to start this problem? I think it is increasing.
| $$5a_{n+2}-5a_{n+1}=2a_{n}-2a_{n+1}$$ or
$$a_{n+2}-a_{n+1}=-\frac{2}{5}(a_{n+1}-a_n),$$
which says that $b_n=a_{n+1}-a_n$ is a geometric progression.
Thus, $$a_{n+1}-a_n=(a_2-a_1)\left(-\frac{2}{5}\right)^{n-1}=\frac{1}{6}\left(-\frac{2}{5}\right)^{n-1}.$$
Thus, $$a_n=a_1+\sum_{k=1}^{n-1}(a_{k+1}-a_k)=\frac{1}{2}+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2652904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Maximum value of $5\sin x - 12 \cos x + 1$ Given that $5\sin x - 12\cos x = 13\sin (x-67.4)$
Find the maximum value of $5\sin x - 12 \cos x + 1 $ and the corresponding value of x from 0 to 360.
Maximum value = $13+1=14$
Corresponding value of $x$
$13\sin (x-67.4) + 1 = 14$
$\sin(x-67.4) = 1 $
$x = 157.4 , 337.4 $ ... | Let $\cos\phi=\dfrac{5}{13}$, $\sin\phi=\dfrac{12}{13}$
\begin{eqnarray}
y&=&13\left(\frac{5}{13}\sin x-\frac{12}{13}\cos x\right)+1\\
&=&13(\sin(x)\cos\phi-\cos(x)\sin\phi)+1\\
&=&13\sin(x-\phi)+1
\end{eqnarray}
which has a maximum value of $14$ when $x-\phi=\frac{\pi}{2}$.
So
\begin{eqnarray}
x&=&\frac{\pi}{2}+\phi\\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Block Matrix Inversion in Wikipedia Wikipedia provides two formulas for block-matrix inversion:
$$ {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}&-... | Provided the inverses involved exist, both terms are equal. But, generally, only one of them is used because the inverse of at least one term involved does not exist. For example, for $\begin{bmatrix}0 & 1\\1 & 1\end{bmatrix}$, the first formula cannot be used, but the second formula can be used to evaluate the inverse... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integer solutions for $x^3=y^2+5$ I have been able to solve the equation $x^3=y^2+a$ for integers where $1 \leq a \leq 4$ by splitting in $\Bbb Z [i\sqrt a]$. However as a natural continuation I would like to know whether the equation $x^3=y^2+5$ can be solved using elementary methods. Seems this case is much tougher t... | Working mod 4 we see that $y$ must be even and $x=1\bmod4$. Now we have $$y^2+4=x^3-1=(x-1)(x^2+x+1)$$ But $x^2+x+1=3\bmod4$, so $x^2+x+1$ is odd. It is also positive, so it must be at least 3. Hence it must have a prime factor $p=3\bmod4$. So $y^2+4=0\bmod p$, in other words -4 is a quadratic residue of $p$. But that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How many ways are there to get a sum of $25$ when $10$ distinct dice are rolled? I have come across the following problem and have given it a good attempt below. I am wondering if I have proceeded correctly, and if not if someone could show me the correct answer or maybe a more efficient solution, thanks!
How many way... | You will enjoy reading about Polynomial Coefficients here. See specifically Theorem 2.1 in the referenced paper on the sum of discrete uniform random variables.
What you are looking for is what's referred to as multinomials. Your approach tries to distill a multinomial into its basic binomial counter parts. There is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Real Analysis. Differentiation (Taylor's Formula). Check proof.
Show that the polynomial
$$x - \frac{x^3}{3!} + \frac{x^5}{5!} - ... \pm \frac{x^n}{n!},$$
with $n$ odd, differs from $\sin x$ by, at most, $\frac{\pi^{n+1}}{(n+1)!}$ in $[-\pi, \pi]$.
$\textbf{Solution:}$ Applying the Taylor's Formula in $f(x) = \si... | Yes, your proof is correct.
I would change $$ R_{n+1} = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1} \leq \frac{\pi^{n+1}}{(n+1)!}.$$
to $$ |R_{n+1}| = \frac{|f^{(n+1)}(c)|}{(n+1)!}x^{n+1} \leq \frac{\pi^{n+1}}{(n+1)!}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Alternate solution to a limit without using L'Hopital's rule $$\lim \limits_{x \to a} \frac{x}{x-a} \left(\frac{x^3}{(a-1)^2}-\frac{a^3}{(x-1)^2}\right)$$
I've gotten to this
$$a \cdot \lim \limits_{x \to a} \frac{x^5-2x^4+x^3-a^5+2a^4-a^3}{(x-a)(a-1)^2(x-1)^2}$$
since as far as I'm concerned it's just a matter of doi... | One way is to solve with Differentiation From First Principles perspective:
$\displaystyle \lim \limits_{x \to a} \frac{x}{x-a} \left(\frac{x^3}{(a-1)^2}-\frac{a^3}{(x-1)^2}\right)=\lim \limits_{x \to a} \frac{x}{x-a} \left(\dfrac{x^3(x-1)^2-a^3 (a-1)^2}{(a-1)^2 (x-1)^2}\right)=\dfrac{a}{(a-1)^4} \left( \lim_{x \to a} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665467",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Compute $\lim_{x\to\frac{\pi}{2}} \frac{(1-\sin x)(1-\sin^2x)\dots(1-\sin^nx)}{\cos^{2n}x}$
$$\lim_{x\to\frac{\pi}{2}} \frac{(1-\sin x)(1-\sin^2x)\dots(1-\sin^nx)}{\cos^{2n}x}=?$$
Here's what I have done so far:
$y=\frac{\pi}{2}-x$ The limit becomes:$$\lim_{y\to 0}\frac{(1-\cos y)(1-\cos^2y)\dots(1-\cos^ny)}{\sin^{2n... | \begin{align}\lim_{x\to\frac\pi2}\frac{(1-\sin x)(1-\sin^2x)\dots\lim(1-\sin^nx)}{\cos^{2n}x}&=\lim_{x\to\frac\pi2}\frac{1-\sin x}{\cos^2x}\cdot\frac{1-\sin^2x}{\cos^2x}\cdot\frac{1-\sin^nx}{\cos^2x}\\&=\frac12\cdot\frac22\cdots\frac n2=\frac{n!}{2^n}\end{align}because, near $\frac\pi2$, $1-\sin^kx$ beahaves like $\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2666056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find the roots of $\mathrm{ \overline{Z} + 1 = iZ^2 + {|Z|}^2}$ I attempted the problem in two ways
Method 1
Resolved $\mathrm{ Z = X + iY}$ this led to a very big cubic equation and handling that became too difficult for me .
Method 2
I believe factorizing the equation is a far better method, I don't know how ... | The complex number $Z$ can be written as $$Z = A + iB.$$
Then, your equation becomes:
$$\overline{Z} + 1 = iZ^2 + |Z|^2 \Rightarrow \\
A-iB + 1 = i(A^2-B^2+2iAB)+A^2+B^2 \Rightarrow \\
A-iB + 1 = iA^2 - iB^2 -2AB + A^2 + B^2 \Rightarrow \\
(A+1 +2AB-A^2-B^2) + i(-B-A^2+B^2) = 0.$$
This means that both the real part $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2668709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Asymptotics of an integral $\mathscr{P} \int_{0}^{\infty} \frac{d\omega}{e^{\omega} - 1} \frac{\omega}{\omega^{2} - x^{2}}$ involving principal value Consider the following integral for $x > 0$:
$$
F(x) \ = \ \mathscr{P} \int_{0}^{\infty} \frac{d\omega}{e^{\omega} - 1} \frac{\omega}{\omega^{2} - x^{2}}
$$
The $\mathscr... | I've figured it out. I've even got an analytic result for this integral!
$$
F(x) := \mathscr{P} \int_{0}^{\infty} \frac{d\omega}{e^{\omega} - 1} \frac{\omega}{\omega^2 - x^2} = \frac{1}{2} \ln\left( \frac{x}{2\pi} \right) - \frac{1}{2} \mathrm{Re}\left[ \psi^{(0)}\left( \frac{ix}{2\pi} \right) \right]
$$
And as we tak... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determine whether $ (A_1,A_2,A_3,A_4)$ span $M_{22}$ Determine whether $\left[\begin{array}{cc}-1&1\\1&-1\end{array}\right], \left[\begin{array}{cc} 1&1\\1&1\end{array}\right], \left[\begin{array}{cc} 1&0\\0&1\end{array}\right],\left[\begin{array}{cc}0&1\\1&0\end{array}\right]$ spans $M_{22}$. If it does, construct the... | Even if the set doesn't span $M_{22}$, it span some subspace $\subseteq M_{22}$; we need to find the basis for such subspace.
To find it you can proceed as for ordinary row/colum vectors by RREF (in this way you could solve also part 1 by this) and selecting matrices corresponding to the pivot rows.
Notably, collecting... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2673729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Easier way to identify a 3 variable determinant as symmetric polynomial How do I identify a 3 variable determinant whether it is a symmetric polynomial or not without actually expanding it ?
Examples:
$$
\begin{vmatrix}
1+a^2&ab&ac\\
ab&1+b^2&bc\\
ca&cb&1+c^2\\
\end{vmatrix}=1+a^2+b^2+c^2
$$
$$\begin{vmatrix}
x+a&y&z\\... | I don't think there's a systematic answer. But a few observations.
(A) I think it is obvious that the examples you give satisfy the $x\to y\to z\to x$ symmetry, so you only need check that swapping $x$ and $y$ is a symmetry. Or if you don't want to do that, ask what would happen if swapping $x$ and $y$ were not a symme... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2674601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of a function tending to a finite number If
$$\lim_{x\to 0} \frac{ae^x - b\cos x +ce^{-x}}{x\sin x} = 2$$
then find the value of $a+b+c$.
My book has given the following solution to the above problem :-
We observe that as $x$ tends to zero , numerator tends to $a-b+c$ whereas the denominator tends to zero. Ther... | Hint. By using Taylor expansions at $0$,
$$\frac{ae^x - b\cos x +ce^{-x}}{x\sin x} =\frac{a(1+x+\frac{x^2}{2}) - b(1-\frac{x^2}{2}) +c(1-x+\frac{x^2}{2})+o(x^2)}{x(x+o(x))}\\
=\frac{(a-b+c)+(a-c)x+\frac{(a+b+c)}{2}x^2+o(x^2)}{x^2+o(x^2)}.$$
In order to have the final limit $2$ we need that
$$(a-b+c)=0,\quad(a-c)=0,\qua... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2676038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Checking my proof of some inequality I would like to receive some help about the next problem:
The Problem:
Prove that it is
$$\frac{1}{n^{1 + \alpha}} < \frac{1}{\alpha} \left(\frac{1}{(n - 1)^\alpha} - \frac{1}{n^\alpha} \right)$$,
if we have that the next is valid:
$$\frac{1}{n^\alpha} - \frac{1}{(n - 1)^\alpha} ... | We can't afford to have $n=0$, as $n$ appears in the denominator. Also, by removing $n=0$, we are sure that $n-\theta >0$ and hence you can raise positive power and preserve the inequality sign in the first line of the solution.
Also, we can't have $n=1$ as $n-1$ appears in the denominator.
Other than that, that is if... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Symmetric Olympiad inequality Prove that for all positive reals $a, b, c$ this inequality holds:
$$\sum_{cyc}\frac{a^3}{b^2 - bc + c^2} \ge \sum_{cyc}a$$
I have proved this in a very ugly way:
I multiplied with $(a^2 - ab + b^2)(b^2 - bc + c^2)(c^2 - ca + a^2)$ and with some work i managed to prove this with Schur. I f... | Using only algebraic manipulation and the fact that a product of similarly-signed numbers is positive:
\begin{gather*}
\sum_\text{cyc} \frac{a^3}{b^2 - bc + c^2} - \sum_\text{cyc} a
= \sum_\text{cyc} \frac{a^3(b + c) - a(b^3 + c^3)}{b^3 + c^3} \\
= \sum_\text{cyc} \frac{ab(a^2 - b^2) - ca(c^2 - a^2)}{b^3 + c^3}
= \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding $\lim_{x\to \infty}(1+\frac{2}{x}+\frac{3}{x^2})^{7x}$ I am having some diffuculty finding the limit for this expression and would appreciate if anyone could give a hint, as to how to continue. I know the limit must be $e^{14}$ (trough an engine) and I can show it for
$(1+\frac{2}{x}+\frac{1}{x^2})^{7x}$ like ... | Note that
$$\left(1+\frac{2}{x}+\frac{3}{x^2}\right)^{7x}=\left[\left(1+ \frac{2}{x}+\frac{3}{x^2}\right)^{\frac1{\left(\frac{2}{x}+\frac{3}{x^2}\right)}}\right]^{{7x}{\left(\frac{2}{x}+\frac{3}{x^2}\right)}}\to e^{14}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Suppose $A \in M_{3,3}$ with eigenvalues $1,2,3$, eigenvectors $b_1, b_2, b_3$. Let $v = b_1 - 4b_2 + 3b_3$. Compute $A^5 v$. I think I did this right, but someone skim and double check?
Q: Suppose $A$ is a $3 \times 3$ matrix, with eigenvalues $1,2,3$ and corresponding eigenvectors $b_1, b_2, b_3$. Suppose that $v = b... | That's too compicated. You might have done just this: Since
*
*$A^5.b_1=b_1$;
*$A^5.(-4b_2)=-4A^5.b_2=-4\times2^5b_2=-128b_2$;
*$A^5.(3b_3)=3A^5.b_3=3\times3^5b_3=729b_3$,
then $A.(b_1-4b_2+3b_3)=b_1-128b_2+729b_3$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Elementary Number Theory: Solving Quadratic Conguences If possible solve the following congruence,
$7x^2-4x+1 \equiv 0 \pmod {11}$.
I am using the quadratic equation but I am stuck. The following is where I got stuck:
$x \equiv 4(4\pm \sqrt{-12}\pmod {11}$
I have tried adding $11$ repeatively but I cannot get a perfec... | So quadratic formula:
$7x^2 -4x + 1\equiv 0 \mod 11\implies$ (using the standard notation abuse)
$x \equiv \frac {4 \pm \sqrt {16 - 28}}{14}\mod 11$
$\frac 12 \equiv 6$ and $\frac 17 \equiv 8$ so $\frac 1{14}\equiv 48 \equiv 4$.
So $x \equiv {16 \pm \sqrt{-12} } \equiv 5\pm \sqrt{-1}$
Now $x^2 \equiv -1 \mod 11$ has no... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the largest cylinder inscribed inside a sphere. Is this calculation correct so far?
A right circular cylinder is inscribed in a sphere of radius $r$. Find the largest possible volume of such a cylinder.
I have that the radius of the cylinder is $r$, the radius of the sphere is $R$, and the height of the inscribe... | You made a sign error in computing the derivative of $\sqrt{R^2-r^2}$.
$$\frac d{dr}\sqrt{R^2-r^2}=\frac{-r}{\sqrt{R^2-r^2}}$$
Thus
$$V'=4\pi r\sqrt{R^2-r^2}-\frac{2\pi r^3}{\sqrt{R^2-r^2}}=0$$
$$4\pi r(R^2-r^2)-2\pi r^3=0$$
$$2(R^2-r^2)=r^2$$
$$2R^2=3r^2$$
$$r=\sqrt{\frac23}R$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2685819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Probability that A wins a best of 7 (4 matches) $A$ and $B$ play a series of best of $4$. The probability that $A$ wins a given game is $p$. Assume that the games are independent of each other. What is the probability that $A$ wins?
Let $A_{i}$ be the event that $A$ wins the series in $i$ matches.
Then, the book states... | I don't see any way to go with your approach without utilizing the book's approach but I will do so anyways.
For $G_5$ to occur, either team $A$ needs to win $3$ or the first $4$ games and then win game $5$ with probability $$\binom{4}{3}p^{4}q$$ or team $B$ needs to win $3$ or the first $4$ games and then win game $5$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
prove that for any nonsingular matrix $A$ there exist $X$ such that $X^2=A$ Prove that given any matrix A, where $$\det(A)\neq0$$ $$A\in M_{n,n}(\mathbb C)$$
the following equation
$$X^2=A$$
always has a solution.
Should I do something with Jordan Normal form?
Any help will be appreciated
| Swear i have done this one fairly recently...
$$
\left(
\begin{array}{rr}
t & \frac{1}{2t} \\
0 & t \\
\end{array}
\right)^2 =
\left(
\begin{array}{rr}
t^2 & 1 \\
0 & t^2
\end{array}
\right)
$$
$$ $$
$$
\left(
\begin{array}{rrr}
t & \frac{1}{2t} & \frac{-1}{8 t^3} \\
0 & t & \frac{1}{2t} \\
0 & 0 & t
\end{array}
\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
I need help with integrating $\int\frac{1}{(x^2-1)^2}dx$ The problem is:
$$\int\frac{1}{(x^2-1)^2}dx$$
I tried using substitution $u=x^2-1$ but that does not bring me got results. I get an integral:
$$\frac{1}{2}\int\frac{1}{u^2\sqrt{u+1}}$$
And that does not really make things any more simple. From here I tried using ... | $$\begin{align*}
\int \frac{1}{(x^2 - 1)^2}\,\mathrm{d}x &\equiv \int \frac{1}{\left((x + 1)\cdot(x - 1)\right)^2}\,\mathrm{d}x \\
&= \int \frac{1}{(x + 1)^2\cdot(x - 1)^2}\,\mathrm{d}x\tag{1}
\end{align*}$$
Decompose $(1)$ into partial fractions,
$$\int \frac{1}{(x - 1)^2\cdot(x + 1)^2}\,\mathrm{d}x = \frac 14\int \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2696618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 0
} |
Find out how many invertible and diagonal solutions $X^2-2X=0 $ has when $ X \in\Bbb{R}^{3\times 3}$ I want to find out how many invertible solutions and how many diagonal solutions matrix equation
has when
$ X \in\Bbb{R}^{3\times 3}.$
I have researched everywhere and can't seem to find the solution to this problem.... | If $X$ is invertible, then multiplying each side of
$$
X^2 - 2X = 0
$$
by the inverse of $X$ gives us that
$$
X - 2I = 0
$$
and so
$$
X = 2I.
$$
For the case where $X$ is diagonal, let
$$
X = \begin{pmatrix} a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c \end{pmatrix}.
$$
Then we have t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2697721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Find all the rational values of $x$ at which $y=\sqrt{x^2+x+3}$ is a rational number Question
Find all the rational values of $x$ at which $y=\sqrt{x^2+x+3}$
My attempt
Since we only have to find the rational values of $x$ and $y$, we can assume that
$$ x \in Q$$
$$ y \in Q$$
$$ y-x \in Q $$
Let$$ d = y-x$$
$$d=\sqrt{... | Hint: the numerator will be $(d^2-d+3)^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2698555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Evaluating Riemann Summation Limit: $\lim\limits_{n\to \infty}\sum\limits_{k=1}^n\frac{6(k-1)^2}{n^3}\sqrt {1+2\frac{(k-1)^3}{n^3}}$ $$\lim_{n\to \infty}\sum_{k=1}^n\frac{6(k-1)^2}{n^3}\sqrt {1+2\frac{(k-1)^3}{n^3}}$$
I have no idea how to approach this problem apart from trying to convert into a definite integral usin... | Hint: By Riemann sum we have
$$\lim_{n\to \infty}\sum_{k=1}^n\frac{6(k-1)^2}{n^3}\sqrt {1+2\frac{(k-1)^3}{n^3}}
\\=6\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n\left(1-\frac{k}{n}\right)^2\sqrt {1+2\left(1-\frac{k}{n}\right)^3}\\=6\int_0^1 (1-x)^2\sqrt{1+2(1-x)^3}dx
\\=6\int_0^1x^2\sqrt{1+2x^3}dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2701164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Throw $4$ red and $4$ black balls into $4$ pots (probability question)
Suppose we have $4$ red and $4$ black balls.
We throw these balls into $4$ pots.
Find the probability:
(a) each pot contains $1$ red and $1$ black ball.
(b) each pot contains $2$ balls.
Attempt. (a) Let $R_i,~B_i$ be the events that pot $i$ c... |
Suppose we have $4$ red and $4$ black balls. We throws these balls into $4$ pots. Find the probability that each pot receives one red and one black ball.
Method 1: Assuming each ball lands in one of the pots, there are four places each of the eight balls could land. Therefore, there are $4^8$ ways to distribute t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prob. 10, Sec. 3.3, in Bartle & Sherbert's INTRO TO REAL ANALYSIS, 4th ed: Is this sequence convergent? Here is Prob. 10, Sec. 3.3, in the book Introduction to Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition:
Establish the convergence or the divergence of the sequence $\left( y_n \right)$, where
... | What you did is fine.
Now, as far as the imit is concerned, note that$$y_n=\frac1n\left(\frac1{1+1/n}+\frac1{1+2/n}+\cdots+\frac1{1+n/n}\right).$$This is a Riemann sum.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Find derivative of $y=\sin^{-1}\frac{2x}{1+x^2}$
Find $\frac{dy}{dx}$ if $y=\sin^{-1}\frac{2x}{1+x^2}$
The solution is given as $\frac{2}{1+x^2}$. But is it a complete solution ?
My Attempt
$$
2\tan^{-1}x=\begin{cases}\sin^{-1}\frac{2x}{1+x^2},\quad |x|\leq 1\\
\pi-\sin^{-1}\frac{2x}{1+x^2},\quad |x|>1 \;\&\; x>0\\
-... | Note that
*
*$(\sin^{-1}x)'=\frac{1}{\sqrt{1-x^2}}$
then apply chain rule $f(g(x))'=f'(g(x))g'(x)=\frac{\frac{2(1+x^2)-4x^2}{(1+x^2)^2}}{\sqrt{1-{\left(\frac{2x}{1+x^2}\right)}^2}}=\frac{2(1-x^2)}{(1+x^2)|1-x^2|}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2705106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
For which positive integers $k$ dividing $4p^3$, does the equation $x^3+px=k^2$ has an integer solution in $x$? Let $p$ be an odd prime. For which positive integers $k$ dividing $4p^3$, does the equation $x^3+px=k^2$ has an integer solution in $x$ ?
Since for integer $x$, $px+x^3$ is even, so $k$ is even, so $4|x^3+px$... | The question as stated is really just 12 different Diophantine equations:
\begin{align*}
&x^3+px=1 & &x^3+px=4 & &x^3+px=16 \\
&x^3+px=p^2 & &x^3+px=4p^2 & &x^3+px=16p^2 \\
&x^3+px=p^4 & &x^3+px=4p^4 & &x^3+px=16p^4 \\
&x^3+px=p^6 & &x^3+px=4p^6 & &x^3+px=16p^6
\end{align*}
As already noted, the LHS of each equation is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2707603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove that the sequence $b_n = 1$, $b_{n+1} = \frac{1}{1+b_n}$ converges. I need to prove that the sequence $b_1 = 1$, $b_{n+1} = \frac{1}{1+b_n}$ converges.
If $\lim_{n\to\infty} b_n = L$ then $\lim_{n\to\infty} b_{n+1} = L$, so $\lim_{n\to\infty} b_{n+1} = \frac{1}{1+\lim_{n\to\infty} b_n} \implies L = \frac{1}{1+L} ... | Here's another interesting way of proving its convergence (And might provide some nice insights into the limit): Define
$$b_n=\frac{h_n}{g_n}$$
where these are the numerator and denominator of $b_n$ (with no reductions to a simpler form after appyling the recursion formula). Then from your definition
$$b_{n+1}=\frac{h_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2708114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving that $ \mid z+1\mid^{2}=2\mid z\mid^{2} \Leftrightarrow \mid z-1\mid^{2}=2 $ Let z be a complex number.
How does one prove, using the properties of the conjugate and the modulus, that:
$$ \mid z+1\mid^{2}=2\mid z\mid^{2}
\Leftrightarrow
\mid z-1\mid^{2}=2 $$
| An elementary solution:
First, note that $|x+iy|^2=x^2+y^2,|x+1+iy|^2=x^2+2x+1+y^2$, and $|z-1 |^2=(x-1)^2+y^2.$
We have then
$$ x^2+2x+1+y^2=2x^2+2y^2$$
or $$x^2-2x+1-2+y^2=0$$
which is equivalent with the following
$$(x-1)^2+y^2=2$$
and again, this is what we needed
$$|z-1 |^2=2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2710156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Last three digits of $6^{2002}$ Find the last three digits of $6^{2002}$. I did some work and figured out that the last two digits is 36. Can anyone help me with the hundredth digit? By the way, I used modular arithmetic and the recursion method for the tens digit, but it fell short when I attempted to do the hundreds ... | $$6=1+5,6^{25n}=(1+5)^{25n}\equiv1\pmod{5^3}$$
$$\implies6^{25n-1}\equiv6^{-1}\equiv21$$
$$6^{25n+2}\equiv6^3(21)\pmod{5^36^3}$$
$$\equiv216\cdot21\pmod{2^35^3}\equiv?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Center manifold theorem example. Does somebody know an easy way to tackle these center manifold problems?
Consider the system
\begin{align}
\dot x&=ax^3+x^2y\\
\dot y &=-y+y^2+xy-x^3
\end{align}
(a) Determine an approximation for the Center Manifold of this system.
(b) Use Center Manifold theory to investigate the... | For small $x$, the first equation tells us that $x$ moves slowly compared to the exponential decay of $y$. Now the (moving) equilibrium of the second equation is at the roots of that quadratic equation, giving roots at $y\approx 1-x$ and $y\approx -x^3$. The first is unstable and far away from $(0,0)$, the second tells... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluate coordinate of turning point : $y={1-\sin x\over 1+\cos x}$ How do I find the coordinate of the turning of $(1)$
$$y={1-\sin x\over 1+\cos x}={u\over v}\tag1$$
Using the Quotient rule:
$${dy\over dx}={vu^{'}-v^{'}u\over v^2}$$
$u=1-\sin x$, $u^{'}=-\cos x$
$v=1+\cos x$, $v^{'}=-\sin x$
$${dy\over dx}={-\cos x(1... | $y={1-\sin x\over 1+\cos x} = \frac{1}{1+\cos{x}}-\frac{\sin{x}}{1+\cos{x}} = \frac{1}{2} \left(1+\tan^2{\frac{x}{2}} \right) - \tan{\frac{x}{2}}$
$y' = \frac{1}{2}\tan{\frac{x}{2}}(1+\tan^2{\frac{x}{2}}) - \frac{1}{2}(1+\tan^2{\frac{x}{2}}) \stackrel{!}{=} 0 \Rightarrow \tan{\frac{x}{2}} = 1 \rightarrow x_t = \frac{\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2714692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Solve: $5\cos^2 (x) - 4\sin (x)\cos (x)+3\sin^2 (x)=2$ Solve to find the general value of $x$:
$$5\cos^2 (x) - 4\sin (x)\cos (x)+3\sin^2 (x)=2$$
My Attempt:
$$5(1-\sin^2 (x))-4\sin (x)\cos (x)+3\sin^2 (x)=2$$
$$5-5\sin^2 (x)-4\sin (x)\cos (x)+3\sin^2 (x)=2$$
$$2\sin^2 (x)+4\sin (x)\cos (x)=3$$
| Guide:
$$2\sin^2(x)+4\sin(x)\cos(x)=3$$
$$2\sin(2x)=2+1-2\sin^2(x)$$
$$2\sin(2x) = 2+\cos(2x)$$
$$2\sin(2x)-\cos(2x)=2$$
The trick from here should help in solving the problem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2724475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Find the horizontal asymptotes of $f(x) = \frac{x^3-2}{\lvert x\rvert^3+1}$ Taken from Thomas' Calculus 12e
Find the horizontal asymptote of the graph of:
$$f(x) = \frac{x^3-2}{\lvert x\rvert^3+1}$$
Solution: We calculate the limits as ${x \to \pm \infty}$
For $x\ge0$: $$\lim\limits_{x\to\infty}\frac{x^3-2}{\lvert x\rv... | I assume the function's domain is $\mathbb R$ except where $f$ is not defined namely $x=-1$.
Now if $\lim_{x \to \infty} f(x) = a$ for some $a \in \mathbb R$, then $y=a$ is a horizontal asymptote.
Similarly, if $\lim_{x \to -\infty} f(x) = b$, for some $b \in \mathbb R$, then $y=b$ is a horizontal asymptote.
For $x\to\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2725683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Show that there are no rational solutions on the ellipse $2x^2 + 3y^2 = 1$ Show that there are no rational solutions on the ellipse $2x^2 + 3y^2 = 1$.
Want to make sure that my proof is correct.
Suppose there are rational solutions.
Note $2x^2 + 3y^2 = 1 \iff 2x^2 + 3y^2 = z^2\ |\ x,y.z \in \mathbb{Z}$
Since taking $(\... | One minor, easily fixed, problem with your proof is that you are using $x$ and $y$ first to denote rational numbers in the equation $2x^2+3y^2=1$, and then to denote the numerators of those rational numbers. It would be better to write $x=a/c$ and $y=b/c$ and then consider the equation $2a^2+3b^2=c^2$ with $a,b,c\in\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2726748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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A closed form for infinite series $ \sum _1 ^\infty \frac {1}{n^3} = \zeta (3) ?$ It is well known that $$ \sum _1 ^\infty \frac {1}{n^2} = \frac {\pi ^2}{6}$$ and $$ \sum _1 ^\infty \frac {1}{n^4} = \frac {\pi ^4}{90}$$
We also know that $$ \sum _1 ^\infty \frac {1}{n^3} $$
$$=1.20205690315959428539973816151144999076... | In addition to robojohn's answer, there are some formulas expressing $\zeta(3)$ (and other odd zeta values) in terms of powers of $\pi$, but these are not clsoed forms. The most well known ones are due to Plouffe and Borwein & Bradley:
$$
\begin{aligned}
\zeta(3)&=\frac{7\pi^3}{180}-2\sum_{n=1}^\infty \frac{1}{n^3(e^{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2727593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proving determinant with variables I have a problem that asks:
Prove that det$\begin{pmatrix}
1 && 1 && 1 \\
a && b && c \\
a^2 && b^2 && c^2 \end{pmatrix} = (c-a)(c-b)(b-a)$
I was thinking of solving it like normal and see if I can end up with $(c-a)(c-b)(b-a)$
What I did:
det$\begin{pmatrix}
b && c \\
b^2 && c^2 \e... | If a=b, b=c, c=a then determinant value is 0 so
(a-b)(b-c)(c-a) are factor of the determinant now comparing the leading coefficient so the Value of the determinant is (a-b)(b-c)(c-a)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2728781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Explaining $\int_{-1}^1\frac{1}{1+x^2}\,dx = \frac{\pi}{2}$. How would you explain to a student that
$$ \int_{-1}^1\frac{1}{1+x^2}\,dx = \arctan(1) - \arctan(-1) = \frac{\pi}{2} $$
and not
$$ \int_{-1}^1\frac{1}{1+x^2}\,dx = \arctan(1) - \arctan(-1) \neq \frac{\pi}{4} - \frac{3\pi}{4} = -\frac{\pi}{2}$$
besides the obv... | Well, you can also use the even function 'card',
i.e, since $f(x) = f(-x)$,
$$ \int_{-1}^{1}\frac{1}{1+x^2}=2\int_{0}^{1}\frac{1}{1+x^2}=2(\arctan 1 +\arctan 0) = 2 \, \left(\frac{\pi}{4} \right) =\frac{\pi}{2} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2729549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
Find Jordan canonical form and basis of a linear operator. Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear operator such that: $T(x,y,z)=(-y-2z,x+3y+z,x+3z)$, I need to find a Jordan canonical form and a basis.
This is what i did:
In the first place, I found the associated matrix to this linear operator in the canon... | Usually the Jordan normal form is as follow
$$J=\begin{pmatrix}
2 & 1 & 0 \\
0 & 2 & 1 \\
0 & 0 & 2 \\
\end{pmatrix}$$
By Jordan theorem we know that a matrix $P$ exists such that $$P^{-1}AP=J$$
let $$P=[v_1,v_2,v_3,v_4]$$
then P has to satisfy the following system: $$AP=PJ$$ that is in this case $$Av_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2730123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the limit of sequence $\frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+\cdots+\frac{2n-1}{2^{n}}$ without using of derivatives and etc. I need to find limit of sequence
$$
\lim_{n \to \infty }\left(\frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+\cdots+\frac{2n-1}{2^{n}}\right)
$$
I tried to solve it and stopped here
$$
f... | $$\lim_{n \to \infty }\left(\frac{1}{2}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+...+\frac{2n-1}{2^{n}}\right)=\sum _1^{\infty} \frac {2n-1}{2^n}=2\sum _1^{\infty} \frac {n}{2^n} -\sum _1^{\infty} \frac {1}{2^n}=2\sum _1^{\infty} \frac {n}{2^n} -1$$
Note that
\begin{align}
\sum _1^{\infty} \frac {n}{2^n}&=\left(\frac12+\frac14+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2732288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
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Solve system:$yz(y+z-x)= a(x+y+z)\\ zx(z+x-y)= b(x+y+z)\\ xy(x+y-z)= c(x+y+z)$ Solve system:
$$yz(y+z-x)= a(x+y+z)\\
zx(z+x-y)= b(x+y+z)\\
xy(x+y-z)= c(x+y+z)$$
$$a, b, c>0$$
I can only solve the system at $a= 2, b= 5, c= 10$. I have tried all things but it's hard with me. Somebody help me?
| Solve equation 1 for $x$:
$$ x=-{\frac { \left( y+z \right) \left( -yz+a \right) }{yz+a}}$$
(unless $yz+a=0$, which I'll ignore for now).
Substitute this into the other two equations, and simplify. I get
$$ \eqalign{yz \left( y+z \right) \left( y{z}^{3}-ayz-a{z}^{2}-byz+{a}^{2}-ab
\right) &= 0\cr
yz \left( y+z \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2733264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Let A be a matrix then $A^{50}$ is?
If $$A=\begin{bmatrix}1&0&0\\1&0&1\\0&1&0\\ \end{bmatrix}$$ then ${A^{50}}$ is?
How to calculate easily? What is the trick behind it? Please tell me.
| Although there are easier ways in this particular case, I will outline a trick involving the Cayley-Hamilton theorem.
We have $\det\begin{pmatrix}
1-X & 0&0\\
1& -X& 1\\
0&1&-X
\end{pmatrix}=-(X+1)(1-X)^2=Q(X)$. Hence there are two eigenvalues, namely $-1$ and $1$. We know that $A$ is diagonalizable if there are two i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2738272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
The square of the solution of the equation... The square of the solution of the equation
$x\sqrt{7} + \sqrt{8 - 3\sqrt{7}} - \sqrt{8 + 3\sqrt{7}} = 0$
is equal to: ...
$x\sqrt{7} + \sqrt{8 - 3\sqrt{7}} - \sqrt{8 + 3\sqrt{7}} = 0$
$\implies x\sqrt{7} = \sqrt{8 + 3\sqrt{7}} - \sqrt{8 - 3\sqrt{7}}$
$\implies 7x^2 = \left... | Note that $$(a-b)^2 = a^2 + b^2 - 2ab $$
Your confusion is the $$-2ab = - \underline{2\sqrt{\left(8+3\sqrt{7}\right)\cdot\left(8-3\sqrt{7}\right)}} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2739227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
For what real values of $x$ does the series $\sum_{n=1}^{\infty} \frac{n}{5^{n-1}}(x+2)^n $ converge?
For what real values of $x$ does the series $\sum_{n=1}^{\infty} \frac{n}{5^{n-1}}(x+2)^n$ converge?
My attempt: $-1< (x+2)^n < 1$. Now, $-3 < x < -1$, so by the Leibniz test, the given series will converge in $-3 ... | HINT
By ratio test
$$\left| \frac{(n+1)(x+2)^{n+1}}{5^{n}}\frac{5^{n-1}}{n(x+2)^n}\right|=\frac{n+1}{5n}\left|x+2\right|\to\frac{1}{5}\left|x+2\right|$$
then the series converges for $|x+2|<5 \implies -7<x<3$.
Then check separetely the cases $x=-7$ and $x=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2739480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding a tight lower bound for $\left(\frac{1+x}{(1+x/2)^2}\right)^n$ I am trying to find a tight lower bound for $\left(\frac{1+x}{(1+x/2)^2}\right)^n$ as a function of $x$ and $n$ and for large $n$,
where $x$ changes with $n$ such that $\lim_{n\to\infty}x=0$.
I am not sure wether my approach to solve this is righ... | Write $y = x/2$.
Then
$\left(\frac{1+x}{(1+x/2)^2}\right)^n
=\left(\frac{1+2y}{(1+y)^2}\right)^n
=\frac{(1+2y)^n}{(1+y)^{2n}}
$.
Since
$(1+2y)^n
=\sum_{j=0}^n \binom{n}{j}2^jy^j
$
and
$\frac1{(1+y)^{2n}}
=\sum_{k=0}^{\infty} \binom{2n+k-1}{k}(-1)^ky^k
$,
$\begin{array}\\
\frac{(1+2y)^n}{(1+y)^{2n}}
&=\sum_{j=0}^n \bino... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Elementary proofs for Taylor expansion for natural logarithm For $|x|<1$, we have that
$$ \ln(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \cdots $$
Is there any elementary(try not to use integration or differantiation) proof for the equality above?
Edit: I have changed my definition of elemantary.
| Here is a sketch, which if extended will find any particular coefficient without calculus, on the assumption a polynomial expression exists
Let's start with the definitions:
*
*$\exp(x)= 1+\frac{1}{1!}x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\cdots$ and
*for positive $x$ we have $\exp(\ln(x))=x$
Suppose $\ln(1+x)=a_0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2741928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Complex integral $I=\int_{|z|=2} \frac{z^3 e^{\frac{1}{z}}}{z+1}dz$ I have this integral from someone who told me it has a nice answer.$$I=\int_{|z|=2} \frac{z^3 e^{\frac{1}{z}}}{z+1}dz$$ I tried to evaluate this using residues theorem, and expanded into series to find the residue at $\infty$$$f(z)=z^3[\sum_{n=0}^{\inf... | Substitute $u = \frac 1z$, so the integral is over the circle $|u| = 1/2$, and is now clockwise. $z = \frac 1u$, so $dz = -\frac{1}{u^2} du$, and so the minus signs cancel
$$\begin{align}I &= \int_{|u| = \frac{1}{2}} \frac{\left(\frac{1}{u}\right)^3 e^u}{\frac{1}{u} + 1} \frac{1}{u^2}\, du\\
&=\int_{|u| = \frac{1}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2743175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find $\lim_{x\rightarrow\frac{\pi}{6}}\frac{1-2\sin x}{\cos3x}$
Find $$\lim_{x\rightarrow\frac{\pi}{6}}\frac{1-2\sin x}{\cos3x}$$
without L'Hôpital's rule.
My work:
1) I know that $$\lim_{x\rightarrow0}\frac{\sin x}{x}=1$$
2) Let $x=t-\frac{\pi}{6}$. Then
$$\lim_{x\rightarrow\frac{\pi}{6}}\frac{1-2\sin x}{\cos3x}=... | Hint:
$1-2\sin(t-\frac{\pi}{6})=1-2(\sin t\cos\frac{\pi}{6}+\cos t\sin\frac{\pi}{6})=2\sin^2\frac{t}{2}-\sqrt{3}\sin t$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2744745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
attempting to solve the diff equation $y'(x^2+1)-2xy=4\sqrt{y(x^2+1)}$ $$y'(x^2+1)-2xy=4\sqrt{y(x^2+1)}$$
$$y'\sqrt{(x^2+1)}-\frac{2xy}{\sqrt{(x^2+1)}}=4\sqrt{y}$$
$$\frac{y'\sqrt{(x^2+1)}-\frac{2xy}{\sqrt{(x^2+1)}}}{x^2+1}= \frac{4\sqrt{y}}{x^2+1}$$
I tried to solve this but I can't get rid of the 2 which would have e... | It's Bernouilli's equation
$$y'(x^2+1)-2xy=4\sqrt{y(x^2+1)}$$
do this rather
$$y'y^{-1/2}(x^2+1)-2xy^{1/2}=4\sqrt{(x^2+1)}$$
$$2(y^{1/2})'(x^2+1)-2x(y^{1/2})=4\sqrt{(x^2+1)}$$
Substitute $z=y^{1/2}$
$$z'(x^2+1)-xz=2\sqrt{(x^2+1)}$$
Now you can solve it easily...
Edit
$$(\frac{y}{x^2+1})'= \frac{4\sqrt{y}}{x^2+1}$$
You... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the Equation of the Ellipse By Completing the Square I have the following equation of an ellipse:
$$5x^2+9y^2+40x=100$$
I need to put it in the form:
$$\frac{(x-h)^2}{m^2} + \frac{(y-k)^2}{n^2} =1 $$
I was trying to complete the square with the coefficients that have an $x$ variable.
$$5x^2 + 40x + 9y^2 = 10... | We have
$$5x^2+9y^2+40x=100\iff 5(x+4)^2+9y^2=180\iff (x+4)^2+\frac{y^2}{\frac59}=36\\\iff\frac{(x+4)^2}{36}+\frac{y^2}{20}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"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.