Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Let $n$ be a positive integer such that $\displaystyle{\frac{3+4+\cdots+3n}{5+6+\cdots+5n} = \frac{4}{11}}$ Let $n$ be a positive integer such that
$$\displaystyle{\frac{3+4+\cdots+3n}{5+6+\cdots+5n} = \frac{4}{11}}$$
then
$$\displaystyle{\frac{2+3+\cdots+2n}{4+5+\cdots+4n}} = \frac{m}{p}.$$
The question further "Is... | $k+(k+1)+...+kn=\frac{k}{2}(n+1)(kn-k+1)$ (just the half of the sum of the first and the last terms times the number of the terms)$=\frac{k}{2}(kn^2+n-(k-1))$
Now, from the first equation we get $n$: $$33(3n^2+n-2)=20(5n^2+n-4)$$ $$n^2-13n-14=0$$ $$(n-14)(n+1)=0$$ $$n>0\Rightarrow n=14$$
And from the second equation we... | {
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"url": "https://math.stackexchange.com/questions/121558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A probability game involving dice Consider two players A and B.
Player A rolls a fair, six-sided die $m$ times and notes the highest number on the upper face out of all of the rolls.
Player B rolls the same die $n$ times and notes the highest number on the upper face out of all of the rolls. (It's clear that as $m$ and... | The following ingredients are enough to determine the probabilities.
$1.$ If you toss $n$ times, the probability that the highest number is $\le k$ is $\frac{k^n}{6^n}$.
$2.$ If you toss $n$ times, the probability that the highest number is equal to $k$ is $\frac{k^n -(k-1)^n}{6^n}$.
In ($1$) and ($2$), $k$ ranges fr... | {
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Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$ Let $a,b \in {\mathbb{Z_+}}$ such that $$a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$$
Prove $a=b$
| If $a > b$ then $\frac{a}{b}>1$ and hence there is an $n$ such that $\left(\frac{a}{b}\right)^n > b$, thus $a^n > b^{n+1}$. This contradicts $a^{4k+1} | b^{4k+2}$. The case $a<b$ works in just the same way.
| {
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Evaluate and prove by induction: $\sum k{n\choose k},\sum \frac{1}{k(k+1)}$
*
*$\displaystyle
0\cdot \binom{n}{0} + 1\cdot \binom{n}{1} + 2\binom{n}{2}+\cdots+(n-1)\cdot \binom{n}{n-1}+n\cdot \binom{n}{n}$
*$\displaystyle\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3}+\frac{1}{3\cdot 4} +\cdots+\frac{1}{(n-1)\cdot n}$
Ho... | 1)
$$S_n=\sum_{k=0}^nk\frac{n!}{k!(n-k)!}=n\sum_{k=1}^n\frac{(n-1)!}{(k-1)!(n-k)!}=n\sum_{k=0}^{n-1}\binom{n-1}k=n2^{n-1}.$$
Then the inductive proof by $S_0=0,S_n-S_{n-1}=T_n$, is a little tedious:
$$S_n-S_{n-1}
=\sum_{k=0}^nk\binom nk-\sum_{k=0}^{n-1}k\binom {n-1}k=n\binom nn+\sum_{k=0}^{n-1}k\left(\binom nk-\binom{n... | {
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Limit: How to Conclude I have difficulty to conclude this limit ....; place of my attempts and results, can anyone help?
tanks in advance
$$\lim_{x\to +a}\, \left(1+6\left(\frac{\sin x}{x^2}\right)^x\cdot\frac{\log(1+10^x)}{x}\right),\quad a=+\infty,\,\,\,0,\,\,\,\,-\infty$$
1):$\,\,\,{a=+\infty}$
$$\lim_{x\to +\infty}... | $\lim_{x\to +a}\, \left(1+6\left(\frac{\sin x}{x^2}\right)^x\cdot\frac{\log(1+10^x)}{x}\right),\quad a=+\infty,\,\,\,0,\,\,\,\,-\infty$
When $a=-\infty$,prove as below
take $x=2k\pi-\frac{\pi}{2}$,($k<0,k\in Z$) ,$(\frac{\sin x}{x^2})^x=(\frac{-1}{(2k\pi-\frac{\pi}{2})^2})^{2k\pi-\frac{\pi}{2}}$,when $k \to -\infty$,
$... | {
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Rearranging a formula, transpose for A2 - I'm lost Given the formula:
$$ q = A_1\sqrt\frac{2gh}{(\frac{A_1}{A_2})^2-1} $$
Transpose for $A_2$
I have tried this problem four times and got a different answer every time, none of which are the answer provided in the book. I would very much appreciate if someone could show ... | $$\frac{q^2}{A_1^2} = \frac{2gh}{\left(\frac{A_1}{A_2}\right)^2-1} $$
$$\Leftrightarrow \frac{q^2}{A_1^2}\left(\left(\frac{A_1}{A_2}\right)^2-1\right) = 2gh $$
$$\Leftrightarrow \frac{q^2}{A_2^2}-\frac{q^2}{A_1^2} = 2gh $$
$$\Leftrightarrow \frac{q^2}{A_2^2}= 2gh+\frac{q^2}{A_1^2} $$
$$\Leftrightarrow \sqrt{\frac{q^2}... | {
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Expressing a triple integral as an iterated integral in 6 different ways How can I express the triple integral (for volume) as an iterated integral in six different ways, where the solid I would like the triple integral for is bounded by the following surfaces?
$$z=0,\; x=0,\; y=2,\; z=y-2x$$
Additional advice apprecia... | You want to integrate over:
$z\le y-2x$
$0\le z$
$0\le x$
$y\le 2$
If you sketch a picture, you'll see that this is a tetrahedron with vertices $(0,0,0)$, $(0,2,0)$, $(1,0,2)$, $(1,2,0)$.
Obviously, you have 6 possible permutations of variables $x$, $y$, $z$.
Let's have a look at the range of these variables. We have $... | {
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How to find $\int_0^{2\pi} |a\cos(x)+b\sin(x)|dx$, where $a^2+b^2=1$ I need help to solve this integral:
$$\int_0^{2\pi} |a\cos(x)+b\sin(x)|dx$$ where $a^2+b^2=1$. I hope someone is able to help me.
| Hint: Let $\phi$ be a number such that $a=\sin\phi$ and $b=\cos\phi$. You want to integrate $|\sin(x+\phi)|$. Maybe make the natural change of variable, or let the geometry guide you to the answer.
Remark: The "trick" above has a number of uses. The idea works even if $a^2+b^2\ne 1$. For suppose that $a^2+b^2\ne 0$. No... | {
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Finding $\lim_{x\to\infty} \frac{(2x-5)^4}{(2x^2+1)(3x^2-2)}$ Finding $$\lim_{x\to\infty} \frac{(2x-5)^4}{(2x^2+1)(3x^2-2)}$$
Do I multiply top & bottom by $\frac{1}{x^2}$ or $\frac{1}{x^4}$? How do I distribute them into the numerator tho?
In the answer given, I think with some typos:
$$... = \lim_{x\to\infty} \frac... | The dominant term is $x^4$ (Imagine that you expanded both the numerator and denominator. What would the highest power of $x$ be?). So, you could either multiply numerator and denominator by $1/x^4$ (see below), or factor as follows:
$$\eqalign{
{(2x-5)^4 \over (2x^2+1)(3x^2-2)}
&={ \bigl(x(2-{5\over x})\bigr)^4\over ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove $n^5 - n$ is divisible by $30$ without reduction How can I prove that prove $n^5 - n$ is divisible by $30$?
I took $n^5 - n$ and got $n(n-1)(n+1)(n^2+1)$
Now, $n(n-1)(n+1)$ is divisible by $6$.
Next I need to show that $n(n-1)(n+1)(n^2+1)$ is divisible by $5$.
My guess is using Fermat's little theorem but ... | $$n^2+1\equiv n^2+5n+6\mod 5$$ and $$n^2+5n+6=(n+2)(n+3).$$
Then
$$(n-1)n(n+1)(n+2)(n+3)$$ is divisible by $5$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
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"answer_id": 6
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Finding the linear combinations of two vectors I am studying for my finals and I'm trying to answer the following question:
Consider the following two vectors in $\mathbb{R}^3$: $a=(1,2,3)$ and $b=(2,3,1)$. Decide
whether it is possible to express the vector $c=(2,4,5)$ as a linear
combination of $a$ and $b$.
I ... | Note that your original linear system appears to be completely wrong for the purposes of solving the problem. What you're after is constants $c_1$ and $c_2$ such that
$$
c_1 \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} + c_2 \begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \\ 5 \end{bmatrix},
$$
or, eq... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 2
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$\int_0^{2\pi}\sin\frac{x}{2}\cos^2\frac{x}{2}\,dx$ $$\int_0^{2\pi}\sin\frac{x}{2}\cos^2\frac{x}{2}\,dx$$
Tried substitution ($u = \cos\frac{x}{2}$), but I get $-\frac{\cos^3\frac{x}{2}}{3}$ ($-\frac{2}{3}$) instead of the correct answer, which is $1\frac{1}{3}$
| $u = \cos \dfrac{x}{2} \Rightarrow du = - \dfrac{1}{2}\sin\dfrac{x}{2} \cdot dx$
Then your integrand becomes $-2 u^2 du$.
Can you take it from here?
| {
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"timestamp": "2023-03-29T00:00:00",
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Integrate $\int{2^{2x}} dx$ Integrate $$\int{2^{2x}} dx$$
How do I do it, at first, I thought I treat 2 as $e$ and I will get something like $\dfrac{1}{2} 2^{2x}$, but according to WolframAlpha its supposed to be $\dfrac{4^x}{\lg{4}}$
| $$ I = \int 2^{2x} \mathrm {d}x \tag{1}$$
Let
$$ \begin{align*}
y &=2^{2x} \hspace{5pt} \\
\Rightarrow \ln y &= 2x \ln 2
\end{align*}
$$
Differentiate both sides
$$
\begin{align*}
\frac{1}{y} \frac{dy}{dx} &= 2 \ln 2 \\
\frac{dy}{y}&=
2 \ln 2 \hspace{4pt}dx\\
&= \ln 2^2 \hspace{4pt} dx
\end{align*}
$$
$$
\begin{alig... | {
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Discrete random variable summation problem. $X$ is a discrete random variable taking on the values $X = 1,3,3^2,3^3,\dots,3^m$ and $f(x) = P(X=x)=\dfrac c x$ for a constant $c$. Find $c$.
Solution:
Since $P(X)=1$, we know that $\dfrac c x=1$, so $c=x$. To find x, we have $x = \sum_0^m 3^m$. Since this series summation ... | The probability that $X=1$ is $\dfrac{c}{1}$, the probability that $X=3$ is $\dfrac{c}{3}$, and so on. It follows that
$$\frac{c}{1}+\frac{c}{3}+\frac{c}{3^2}+\cdots +\frac{c}{3^m}=1.$$
We want to find the sum of the finite geometric series
$1+x+x^2+\cdots+x^m$.
where $x=1/3$. Let
$$f(m)=1+x+x^2+\cdots+x^m.$$
Note... | {
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"timestamp": "2023-03-29T00:00:00",
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System of equations - what are the values of the coefficients of a quadratic parabola? In the process of relearning the mathematical basics I'm stumbling over this problem:
A quadratic parabola $y = ax^2 + bx + c $ goes through the points A(1/2), B(3/7) and C(-1/1). What are the values of the coefficients $a$, $b$ and... | First use the fact that the curve passes through the point $(1,2)$. That says that when $x=1$, we have $y=2$. So substitute $x=1$ in the equation $y=ax^2+bx+c$. We get
$$2=a+b+c.$$
Similarly, because $(3,7)$ is on the curve, we have
$$7=9a+3b+c.$$
And finally, the third point tells us that
$$1=a-b+c.$$
We now have $3$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving $a = b^2 + 2b^2(1 - b)$ for $b$ My algebra is very rusty, it's been about 15years since I studied, and I was stumped recently when trying to rearrange this formula;
$$a = b^2 + 2b^2(1 - b)$$
to give $b$ in terms of $a$.
Can someone show me step by step working please :)
I remember 'change side, change sign' but... | Your last equation is correct.
$$\begin{equation*}
a+2b^{3}-3b^{2}=0
\end{equation*}\tag{0}$$
This equation is a cubic equation in $b$, which can be solved algebraically by the Cardano's cubic formula (see e.g. this PlanetMath entry). We can write it as
$$\begin{equation*}
b^{3}-\frac{3}{2}b^{2}+\frac{a}{2}=0.
\end{equ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating $\int_{-20}^{20}\sqrt{2+t^2}\,dt$ I have this integral:
$$\int_{-20}^{20}\sqrt{2+t^2}\,dt$$
I tried solving it many times but without success.
The end result is this:
$$2\left( 10\sqrt{402}+\mathop{\mathrm{arcsinh}}(10\sqrt{2})\right).$$
I can't seem to get this end result. I got a few wrong ones but cant fi... | Another way integrate $\sqrt{2 + t^2}$ is to use the Euler substitution $u = \sqrt{2+t^2} + t$. Subtracting $t$ and squaring gives $t = \frac{1}{2}(u - \frac{2}{u})$. This implies
$$\sqrt{2+t^2} = u - t = \frac{1}{2}(u + \frac{2}{u})$$
and
$$dt = \frac{1}{2}(1 + \frac{2}{u^2})\,du$$
Thus
\begin{align*}
\int \sqrt{2... | {
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"timestamp": "2023-03-29T00:00:00",
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Integrating $\int_{a}^{+\infty} y \exp(-by)/(1-\exp(-cy)) \ dy $ Can any one help me calculate this integral :
$$\int_{a}^{+\infty} \frac{y\ \exp{(-by)}} {1-\exp{(-cy)}} \ dy $$
a, b & c are real constant numbers, b & c > 0
| Assuming $a>0$, you can write the integrand as $\sum_{n=0}^\infty y \exp((-b-nc)y)$ and the integral becomes the convergent series
$$\sum_{n=0}^\infty e^{-ab-acn} \frac{ab+acn+1}{(b+nc)^2} $$
According to Maple this can be written using a hypergeometric function:
$$ \frac{ab+1}{b^2 e^{ab}}\
{\mbox{$_4$F$_3$}(1,{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/147962",
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Finding square roots of $\sqrt 3 +3i$ I was reading an example, where it is calculating the square roots of $\sqrt 3 +3i$.
$w=\sqrt 3 +3i=2\sqrt 3\left(\frac{1}{2}+\frac{1}{2}\sqrt3i\right)\\=2\sqrt 3(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3})$
Let $z^2=w \Rightarrow r^2(\cos(2\theta)+i\sin(2\theta))=2\sqrt 3(\cos\frac{\pi}... | The general procedure (for square roots) goes as follows.
We want the square roots of $a+ib$ (where at least one of $a$ or $b$ is non-zero).
Note that
$$a+ib=\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}+i\frac{b}{\sqrt{a^2+b^2}}i\right).\tag{$1$}$$
For brevity, write $x=\frac{a}{\sqrt{a^2+b^2}}$ and $y=\frac{b}{\sqrt{a... | {
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"timestamp": "2023-03-29T00:00:00",
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calculate the volume of a zone a and b are positive numbers.
Let W be a zone between the surface $$z=2ax + 2by$$ and the parabolic $$z=x^2 + y^2.$$
I need to show that $$\mu(W)=1/2*\pi*R^4$$
I'm not really sure how to begin this.
Would appreciate your help.
| Find the intersection between the plane and paraboloid:
$$
z = x^2 + y^2 = 2ax + 2by
$$
Or
\begin{align*}
x^2 - 2ax + y^2 - 2by &= 0 \\
(x^2 -2ax + a^2) + (y^2 - 2by + b^2) &= a^2 + b^2 \\
(x - a)^2 + (y - b)^2 &= a^2 + b^2
\end{align*}
In the $xy$ plane, this is a disk centered at $(a, b)$ with $R = \sqrt{a^2 + b^2}$:... | {
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How to solve this equation, when the unknown variable just disappears? This problem
$\sqrt{1-x^2} + \sqrt{3+x^2} = 2$
has the solution $x = 1$ and $x = -1$.
However, I always get stuck like this:
*
*$1-x^2 + 3+x^2 = 4$
*$4 = 4$
How do I isolate that darn unknown?
| Square both sides of original equation: $$1-x^2+3+x^2+2\sqrt{(1-x^2)(3+x^2)}=4\Longrightarrow\sqrt{(1-x^2)(3+x^2)}=0\Longrightarrow x =\pm1$$ since $\,3+x^2=0\,$has no real solutions
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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limit of power of fraction of sums of sines Find the following limit:
$$\lim_{n\to\infty} \left(\frac{{\sin\frac{2}{2n}+\sin\frac{4}{2n}+\cdot \cdot \cdot+\sin\frac{2n}{2n}}}{{\sin\frac{1}{2n}+\sin\frac{3}{2n}+\cdot \cdot \cdot+\sin\frac{2n-1}{2n}}}\right)^{n}$$
I thought of some $\sin(x)$ approximation formula, but it... | $$\sum_{k=1}^n \sin \left(a + (k-1)d \right) = \dfrac{\sin(dn/2) \sin(a+d(n-1)/2)}{\sin(d/2)}$$
In your case, for the numerator $a = \dfrac{2}{2n}$ and $d = \dfrac{2}{2n}$. Hence, the numerator is $$ \dfrac{\sin(1/2) \sin(2/2n+2/2n \times (n-1)/2)}{\sin(1/2n)} = \dfrac{\sin(1/2) \sin((n+1)/(2n))}{\sin(1/2n)}$$
Similarl... | {
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"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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simplify $ (-2 + 2\sqrt3i)^{\frac{3}{2}} $? How can I simplify $ (-2 + 2\sqrt3i)^{\frac{3}{2}} $ to rectangular form $z = a+bi$?
(Note: Wolfram Alpha says the answer is $z=-8$. My professor says the answer is $z=\pm8$.)
I've tried to figure this out for a couple hours now, but I'm getting nowhere.
Any help is much app... | So this can be written as $$4^{3/2} \cdot (-\frac{1}{2} + \frac{\sqrt{3}}{2} i)^{3/2} = 8 \cdot (\cos\frac{2\pi}{3} + i\cdot \sin\frac{2\pi}{3})^{3/2}$$
and $4^{3/2} =8$. Use De moivre now. And you can also pull out $-4$ and get going. Hence your $z = \pm{8}$.
| {
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Summation of $ \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \cdots$ till $n$ terms What is the pattern in the following?
*
*Sum to $n$ terms of the series: $$ \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \cdots$$
| Hint:
Write it as $(1-{1\over2})+(1-{1\over4})+(1-{1\over 8})+\cdots+(1-{1\over 2^n}).$
| {
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Solving $x^4-y^4=z^2$ I have a question
show that $x^4- y^4 = z^2$ has no nontrivial solution where $x$, $y$ and $z$ are nonzero integers
I tried infinite descent to find solution but I could not find it. square of a number in mod 4 is 1 or 0 I also tried to use but got nothing.
Can you help?
thanks
| Suppose $z^2=y^4-x^4$ with $xyz\not=0$ for the smallest possible value of $y^4$. First we rewrite the
equation as $y^4=x^4+z^2$ so that $\{z,x^2,y^2\}$ is a Pythagorean triple.
It must be primitive, since if some prime $p$ divides $\gcd(x^2,y^2)$, then
$p\,|\,y^2$ implies $p\,|\,y$ which gives $p^4\,|\,y^4$.
Similarly,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Diophantine equation $a^2+b^2=c^2+d^2$ I was reasonably certain I've seen this before, but I was wondering how to solve the Diophantine equation
$$a^2+b^2=c^2+d^2$$
I tried a web search and found nothing on this one. I'm trying to avoid another library trip to a less than local library (maybe I should have taken bette... | This equation is quite symmetrical so formulas making too much can be written: So for the equation:
$X^2+Y^2=Z^2+R^2$
solution:
$X=a(p^2+s^2)$
$Y=b(p^2+s^2)$
$Z=a(p^2-s^2)+2psb$
$R=2psa+(s^2-p^2)b$
solution:
$X=p^2-2(a-2b)ps+(2a^2-4ab+3b^2)s^2$
$Y=2p^2-4(a-b)ps+(4a^2-6ab+2b^2)s^2$
$Z=2p^2-2(a-2b)ps+2(b^2-a^2)s^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 9,
"answer_id": 2
} |
Polynomials in Fourier trigonometric series I'm successively integrating $x^{n} \cos{k x}$ for increasing values of positive integer n. I'm finding:
$\frac{\sin{kx}}{k}$,
$\frac{\cos{kx}}{k^2}+\frac{x\sin{kx}}{k}$,
$\frac{2 x \cos{kx}}{k^2}+\frac{\left(-2+k^2 x^2\right)sin{kx}}{k^3}$,
$\frac{3 \left(-2+k^2 x^2\righ... | You could also consider the exponential generating function
$$ \sum_{n=0}^\infty \dfrac{t^n}{n!} \int_0^x s^n e^{iks}\ ds = \int_0^x e^{(t+ik)s}\ ds = \dfrac{e^{(t+ik)x} - 1}{t+ik}$$
This is the product of $e^{(t+ik)x}-1 = -1 + \sum_{n=0}^\infty \dfrac{t^n x^n}{n!} e^{ikx}$ and $\dfrac{1}{t+ik} = \sum_{n=0}^\infty (-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
tough series involving digamma I ran across a series that is rather challenging. For kicks I ran it through Maple and it gave me a conglomeration involving digamma. Mathematica gave a solution in terms of Lerch Transcendent, which was worse yet. Perhaps residues would be a better method?.
But, it is $$\sum_{k=1}^{\in... | First, group the consecutive oscillating terms together:
$$\sum_{k=1}^{\infty}\frac{(-1)^{k}(k+1)}{(2k+1)^{2}-a^{2}}=\sum_{k=0}^\infty \left(\frac{2k+1}{(4k+1)^2-a^2}-\frac{2k+2}{(4k+3)^2-a^2}\right)-\frac{2(0)+1}{(2(0)+1)^2-a^2}$$
Next, invoke partial fraction decomposition and solve for coefficients:
$$
\frac{2k+1}{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/153845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Simplify this expression with nested radical signs My question is-
Simplify:
$$\frac1{\sqrt{12-2\sqrt{35}}}-\frac2{\sqrt{10+2\sqrt{21}}}-\frac1{\sqrt{8+2\sqrt{15}}}$$
| Let's follow this elementary way to work out things:
$$\frac1{\sqrt{12-2\sqrt{35}}}-\frac2{\sqrt{10+2\sqrt{21}}}-\frac1{\sqrt{8+2\sqrt{15}}}= \frac1{\sqrt{({\sqrt{7}-\sqrt{5}})^2}}-\frac2{\sqrt{({\sqrt{7}+\sqrt{3}})^2}}-\frac1{\sqrt{({\sqrt{5}+\sqrt{3}})^2}}=\frac1{({\sqrt{7}-\sqrt{5}})}-\frac2{({\sqrt{7}+\sqrt{3}})}-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
What is wrong with my solution? $\int \cos^2 x \tan^3x dx$ I am trying to do this problem completely on my own but I can not get a proper answer for some reason
$$\begin{align}
\int \cos^2 x \tan^3x dx
&=\int \cos^2 x \frac{ \sin^3 x}{ \cos^3 x}dx\\
&=\int \frac{ \cos^2 x\sin^3 x}{ \cos^3 x}dx\\
&=\int \frac{ \sin^3 ... | A simple $u$-substitution will work even better, IMO. ${\cos^2\theta\tan^3\theta }$ yields ${\frac {\sin^3}{\cos\theta}}$ So starting from there $${\int\frac{\sin^2\theta}{\cos\theta}\sin\theta d\theta}$$ then $${\int\frac{1-\cos^2\theta}{\cos\theta}\sin\theta d\theta}$$ Let
$u={\cos \theta}$ then ${du=-\sin\theta d\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
"answer_id": 6
} |
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$ Compute
$$\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$$
| Note
\begin{align}
&\int_0^1\frac{\ln (x+1)}{x^2+1}dx
=\int_0^1\underset{x\to\frac{1-x}{1+x}}{\frac{\ln \frac{x+1}{\sqrt{x^2+1}}}{x^2+1}}dx
+ \int_0^1\frac{\ln \sqrt{x^2+1}}{x^2+1}dx\\
&= \int_0^1\frac{\ln \frac{\sqrt2}{\sqrt{x^2+1}}}{x^2+1}dx
+ \int_0^1\frac{\ln \sqrt{x^2+1}}{x^2+1}dx = \frac{\ln2}2\int_0^1\frac{dx}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/155941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "102",
"answer_count": 8,
"answer_id": 4
} |
Prove that $\frac{{a}^{2}}{b-1}+\frac{{b}^{2}}{a-1}\geq8$ I need to prove that for any real number $a>1$ and $b>1$ the following inequality is true:
$$\frac{{a}^{2}}{b-1}+\frac{{b}^{2}}{a-1}\geq8$$
| $$\frac{{a}^{2}}{b-1}+\frac{{b}^{2}}{a-1}=\frac{a^3-a^2+b^3-b^2}{(a-1)(b-1)}=\frac{(a+b)(a^2+b^2-ab)-(a^2+b^2)}{(a-1)(b-1)}\ge\frac{(a+b)ab-(a^2+b^2)}{(a-1)(b-1)}=\frac{a^2(b-1)+b^2(a-1)}{(a-1)(b-1)}=\frac{a^2}{a-1}+\frac{b^2}{b-1}=\frac{a^2-1+1}{a-1}+\frac{b^2-1+1}{b-1}=a+1+\frac{1}{a-1}+b+1+\frac{1}{b-1}=a-1+\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/157688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 1
} |
Linear transformation for projection of a point on a line This is what my textbook wants me to do:
The matrix of the linear transformation $P_L$ that projects $\mathbb{R}^2$ on de straight line $l \leftrightarrow y = mx$ is:
\begin{pmatrix}
\frac{1}{1+m^2} & \frac{m}{1+m^2} \\
\frac{m}{1+m^2} & \frac{m^2}{1+m^2} \\
\e... | You are off to a great start!
If you substitute your second equation into your first, you find $$mx=-\frac{x}{m}+\frac{x_A}{m}+y_A,$$ so $$\frac{1+m^2}{m}x=\frac{x_A}{m}+y_A,$$ and so $$x=\frac{1}{1+m^2}x_A+\frac{m}{1+m^2}y_A,\tag{$\star$}$$ which corresponds to what your first row should be.
Once you've found that, us... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/159603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Plotting a complex argument arc I am having trouble sketching a complex argument arc
$$ \text{Sketch the following on an arcand diagram:}\\ \arg\left(\frac{w+1}{w}\right)=\frac{\pi}{6}$$
I've tried to devise a method on my own looking at questions and answers but it has failed me on this specific question so I requir... | Taking $w = x+iy$, we get that $$\text{Arg} \left( \dfrac{1+w}w\right) = \text{Arg} \left( 1+\dfrac1w\right) = \text{Arg} \left( 1+\dfrac{x-iy}{x^2+y^2}\right) = \text{Arg} \left( \dfrac{x^2+y^2+x}{x^2+y^2} - i \dfrac{y}{x^2+y^2}\right)$$
Hence, we need $$\tan(\pi/6) = -\dfrac{y}{x^2+y^2+x}$$
$$x^2 + y^2 + x + y\sqrt{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to get $ \cot(\theta/2)$ from $ \frac {\sin \theta} {1 - \cos \theta} $? According to wolfram alpha, $\dfrac {\sin \theta} {1 - \cos \theta} = \cot \left(\dfrac{\theta}{2} \right)$.
But how would you get to $\cot \left(\dfrac{\theta}{2} \right)$ if you're given $\dfrac {\sin \theta} {1 - \cos \theta}$?
| It may not seem obvious at first, but substitute $\theta = 2\varphi$ first because we want to work with integer multiples of the angle rather than fractional multiples.
$$\begin{array}{rcl}
\dfrac{\sin \theta}{1 - \cos \theta} &=& \dfrac{\sin 2\varphi}{1 - \cos 2\varphi}
\end{array}$$
Then we use double angle formula.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/160982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
How do I differentiate this? (logarithm & chain) I keep getting wrong results when trying to differentiate this:
${\partial \over \partial x} \ln{(x - \sqrt{x^2+a^2})}$
Thanks for hints!
| An other method: $y=\ln |x- \sqrt{x^2+a^2}|$ gives :
$$\begin{align*} e^y &= x-\sqrt{x^2+a^2}\\ (e^y-x)^2 &=x^2+a^2 \\ 2(e^y-x) \left(e^y\frac{dy}{dx} -1\right) &=2x \\ e^y\frac{dy}{dx}&= 1+\frac{x}{e^y-x}=\frac{e^y}{e^y-x}\end{align*}$$
Since : $e^y-x=-\sqrt{x^2+a^2}$, this gives :
$$\frac{dy}{dx}= - \frac{1}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/161039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
If $ \frac{1}{c} = \frac{1}{2a} + \frac{1}{2b} $, then either $ a \leqslant c \leqslant b$ or $ b \leqslant c \leqslant a $ For $a, b = 1, 2, 3, \cdots$, let $ \frac{1}{c} = \frac{1}{2a} + \frac{1}{2b} $. Then prove that either $ a \leqslant c \leqslant b$ or $ b \leqslant c \leqslant a $ holds.
| $c\geq a\Rightarrow 1/2a+1/2b\leq 1/a\Rightarrow1/2b\leq1/2a\Rightarrow 1/2a+1/2b\geq1/b\Rightarrow c\leq b$. Similarly do when $c\leq a$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/162096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Parametric equations and area I am not sure how to work this one out.
I am suppose to find the area of this parametric equation.
$$y = b\sin\theta, x = a\cos\theta$$
$$0 \leq 0 \leq 2\pi$$
I set up the equation in the memorized formula.
$$\int_0^{2\pi} \sqrt{1 + \left(\frac{b\cos\theta}{-a\sin\theta}\right)^2}d\theta$$... | Note that $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=\cos^2\theta + \sin^2\theta=1.$ We then recognize that this describes an ellipse, which has area $\pi a b$.
EDIT: How to find the area of the ellipse.
There are many many proofs of this, but the easiest one you might find in a single-variable calculus cou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Factoring multivariate polynomial I'm trying to factor
$$x^3+x^2y-x^2+2xy+y^2-2x-2y \in \mathbb{Q}[x,y].$$
The hint for the exercise is to use the recursive multivariate polynomial form. So I'm using $\mathbb{Q}[x][y]$:
$$ x^3 + x^2(y-1) + x^1(y-2) + x^0(y^2-2y) $$
At this point I am stuck. Are their any general techni... | Let’s go back to the original polynomial,
$$x^3+x^2y-x^2+2xy+y^2-2x-2y\;.\tag{1}$$
That $2xy$ looks a lot like the middle term of $(x+y)^2$, and the $-2x-2y$ can certainly be written nicely in terms of $x+y$, so let’s try something along those lines. $(x+y)^2=x^2+2xy+y^2$, and we have the $2xy+y^2$, but instead of $x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/163417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find the value of $(a^3 + b^3 + c^3)/(abc)$ if $a/b + b/c + c/a = 1$.
Find the value of $$\frac{a^3+b^3+c^3}{abc}\qquad\text{ if }\quad \frac ab + \frac bc + \frac ca = 1.$$
I tried using Cauchy's inequality but it was of no help. Please guide me.
$a, b, c$ are real.
| Let
$$\begin{eqnarray}
u&=&a+b+c\\
v&=&a^3+b^3+c^3\\
s&=&a^2b + b^2 c + c^2 a\\
t &=& a \, b\, c
\end{eqnarray}$$
These are related by
$$u^3 = v + 3 s + t $$
Further, we are given
$$ \frac ab + \frac bc + \frac ca = \frac st =1$$
so $$u^3 = v + 4 t$$
And our target is
$$\frac{a^3+b^3+c^3}{abc}=\frac vt = \frac{u^3}{t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/164623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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$X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$ $X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$
I changed the bases to 10, then performed manual addition/multiplication but it didn't yield me any result except for long terms. Please show me the way.
All I'm getting is $$\frac{\lg 18\lg54 +... | Let $I=\dfrac{\log 18}{\log 12}\cdot\dfrac{\log 54}{\log 24}+5 \left( \dfrac{\log 18}{\log 12}-\dfrac{\log 54}{\log 24} \right)$. Also, let $\log 3=x $ and $\log 2=y$.
Then,
$$I= \frac{\log 3^2\cdot2}{\log 2^2\cdot 3}.\frac{\log 3^3 \cdot 2}{\log 2^3\cdot 3}+5\left(\frac{\log 3^2\cdot2}{\log 2^2\cdot3}-\frac{\log 3^3\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/165667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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All integer solutions for $x^4-y^4=15$ I'm trying to find all the integer solutions for $x^4-y^4=15$.
I know that the options are $x^2-y^2=5, x^2+y^2=3$, or $x^2-y^2=1, x^2+y^2=15$, or $x^2-y^2=15, x^2+y^2=1$, and the last one $x^2-y^2=3, x^2+y^2=5$.
Only the last one is valid. $x^2+y^2=15$ is not solvable since the p... | $$(x^2-y^2)+(x^2+y^2)=2x^2=3+5=8\Rightarrow x^2=4\Rightarrow x=2,-2$$
$$(x^2+y^2)-(x^2-y^2)=2y^2=5-3=2\Rightarrow y^2=1\Rightarrow y=1,-1$$
Substitute the values to check that these are indeed the soloutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/166070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
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Complex part of a contour integration not using contour integration A propos of a user's comment on this question, quoting Feynman to the effect that some integrals are only possible using contour integration, I wonder what the simplest example of such an integral might be. In particular, he spoke of integrals that wer... | A change of variables yields
$$
\begin{align}
\int_0^\infty\frac{\log(x)^2}{1+x^2}\,\mathrm{d}x
&=\int_{-\infty}^\infty\frac{t^2}{1+e^{-2t}}e^-t\,\mathrm{d}t\\
&=2\int_0^\infty\frac{t^2}{1+e^{-2t}}e^{-t}\,\mathrm{d}t\\
&=2\int_{-\infty}^\infty t^2\left(e^{-t}-e^{-3t}+e^{-5t}-e^{-7t}+\dots\right)\,\mathrm{d}t\\
&=4\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/167304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Finding the area of triangle if length of medians are given My question is:
In triangle ABC ,length of median from vertex A is $13$ , length of
median from vertex B is $14$ , length of median from vertex C is $15$.
Compute the area of triangle ABC.
| One strategy, probably not optimal, is to find the lengths of the sides, and then use Heron's Formula.
Take a triangle $XYZ$, with sides $x$, $y$, $z$ as usual, and let $m$ be the length of the median from $Z$ to the side $XY$, which has length $z$.
Let $P$ be the midpoint of $XY$. We have divided our triangle into tw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/168701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Finding all primes $p$ such that $\frac{(11^{p-1}-1)}{p}$ is a perfect square How to Find all primes $p$ such that $\dfrac{(11^{p-1}-1)}{p}$ is a perfect square
| Let $\frac{a^{p-1}-1}{p}=b^2$ for some integer b.
If p=2, $a=2b^2+1$(11 is not of the form).
If p>2 & prime ,so must be odd=(2k+1) (say)
$(a^k+1)(a^k-1)=b^2(2k+1)$
Now, if a is odd (like 11), $a^k±1$ is even => b is even =2d(say).
$\frac{a^k+1}{2}\frac{a^k-1}{2}=d^2(2k+1)$
Now, $(\frac{a^k+1}{2},\frac{a^k-1}{2})$=1
So,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/169935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 0
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Induction Proof: $5^n + 5 < 5^{n+1}$ I am trying to prove for all natural $n$ that:
$$5^n + 5 < 5^{n+1}$$
I did the basic step with $n=1$ and inequality holds, I am now at the induction step:
$$5^{k+1} + 5 < 5^{k+2}$$
and I have no idea how to proceed from here. Can someone give me a clue?
| Inequality holds for $n=1$.
For $n=k$
$$5^k + 5 < 5^{k+1}$$
We can write this as
$$5^{k+1} - (5^k + 5) = p$$
where $p$ is a positive integer.
$$5*5^k - 5^k - 5 = p$$
$$4*5^k - 5 = p$$
$$5^k=\frac{p+5}{4}$$
Now we have to prove it is true for $n = k+1$. So if $(5^{k+2} - (5^{k+1}+5))$ is positive then inequality holds t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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How to solve this Pell's equation $x^{2} - 991y^{2} = 1 $ How to solve the following Pell's equation?
$$x^{2} - 991y^{2} = 1 $$
where $(x, y)$ are naturals.
The answer is $$x = 379,516,400,906,811,930,638,014,896,080$$
$$y = 12,055,735,790,331,359,447,442,538,767$$
I can't think of any way apart from brute force. Plea... | $$x^2 - 991y^2 = 1$$ -------(1)
Assuming $y \not= 0$ divide both sides of the equation by $y^2$
i.e. $$(x/y)^2 - 991 = (1/y)^2$$
i.e. $(x/y)^2 - (1/y)^2 = 991$
i.e. $(x/y - 1/y)(x/y + 1/y) = (1)(991)$
i.e. $(x/y - 1/y)(x/y + 1/y) = (496 - 495)(496 + 495)$
i.e. $x/y = 496$ and $1/y = 495$
i.e. $x = (496/495)$ and $y = 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 4
} |
Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$ Numerically it seems to be true that
$$
\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}.
$$
Any ideas how to prove this?
| Let's start out with the following relation:
$$\int_0^{\infty} \frac{\sin x}{\sqrt{x}} e^{-a x} dx = \frac{2}{\sqrt{\pi}} \int_0^{\infty} \frac{1}{1+(a+x^2)^2} dx \tag1$$
Proof of the relation $(1)$
$$\int_0^{\infty} \frac{\sin x}{\sqrt{x}} e^{-a x} dx=$$
Notice that $\displaystyle \frac{1}{\sqrt x}= \frac{2}{\sqrt{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/171970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 8,
"answer_id": 1
} |
Evaluating $\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$ How do I integrate this expression:
$$\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$$.I got this in a book.I do not know how to evaluate integrals of this type.
| One uses trigonometric substitution: $t = \tan\left(\frac{x}{2}\right)$. Then
$$
\sin(x) = \frac{2t}{1+t^2} \quad \cos(x) = \frac{1-t^2}{1+t^2} \quad \mathrm{d} x = \frac{2}{1+t^2} \mathrm{d} t
$$
Hence:
$$\begin{eqnarray}
\int \frac{ \ell \sin(x) + m \cos(x)}{(a \sin(x)+ b \cos(x))^2} \mathrm{d}x &=& \int \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/172698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Find the remainder for $(x^5-x^4+x^3+2x^2-x+4)/(x^3+1)$
Find the remainder for $\dfrac{x^5-x^4+x^3+2x^2-x+4}{x^3+1}$
I know exactly how to synthetically divide in the format of: $(x\pm a)$. But not $(x^n\pm a)$ (with an exponent). So if anyone can tell me if anything changes or if the steps are the exact same just ... | All you have to do is do long division; alternatively, you can subtract appropriate multiples of $x^3+1$ and replace the dividend by the difference until you get a term that is of degree strictly smaller than $3$. (In other words, do polynomial long division analytically instead of synthetically).
For instance, suppose... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Solve the following inequality $x^2+x+1\gt 0$
Solve the following inequality $x^2+x+1\gt 0$
I understand how to solve inequalities and what the graphs look like. Usually the first step is to set this as in equation and then find the zeros. But for this one when I used the quadratic formula my two answers were:
$$\df... | Complete the square: $x^2+x+1=\left(x+\frac12\right)^2+\frac34$. For what values of $x$ is this positive?
Added: Here’s an explanation of completing the square. Suppose that you have a quadratic $x^2+ax+b$. You want to write this in the form $(x+c)^2+d$ for some constants $c$ and $d$. We know that $(x+c)^2+d=x^2+2cx+(c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/173140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Laurent series of $f(z)=\frac{1}{z(z-1)(z-2)}.$ Consider $$f(z)=\frac{1}{z(z-1)(z-2)}.$$
I want to determine the Laurent series in the point $z_0=0$ on $0<|z|<1$.
Partial decomposition yields:
$$f(z)=\frac{1}{z(z-1)(z-2)}=(1/2)\cdot (1/z) - (1/(z-1)) + (1/2)(1/(z-2)).$$
Is the general strategy now, to try to use the ge... | Partial fractions and geometric series give
$$
\begin{align}
\frac1{(1-x)(2-x)}
&=\frac1{1-x}-\frac1{2-x}\\
&=\frac1{1-x}-\frac12\frac1{1-x/2}\\
&=(1+x+x^2+x^3+x^4+\dots)\\
&-\left(\frac12+\frac14x+\frac18x^2+\frac1{16}x^3+\frac1{32x^4}+\dots\right)\\
&=\frac12+\frac34x+\frac78x^2+\frac{15}{16}x^3+\frac{31}{32}x^4+\dot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/174452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solve $x^2 (a - bx) {d^2y \over dx^2} - x (5a - 4bx) {dy \over dx} + 3(2a - bx)y = 6a^2$ What do I substitute here
$$ x^2 (a - bx) {d^2y \over dx^2} - x (5a - 4bx) {dy \over dx} + 3(2a - bx)y = 6a^2 $$ to get it into this form?
$$ u^2 {dv^2 \over du^2} + P_1 u {dv \over du} + P_2 v = F(u) $$
The solution (according t... | I will post a partial answer, as I might have made a sign error or been taken astray at some point. The method amounts to finding some sort of integrating factor for the equation. It seems reasonable to start with the change of the dependent variable:
$$u=a-bx$$
$$x=-\frac{1}{b}\left(u-a\right)$$
$$\frac{dy}{dx}=-b\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/176220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove $\sin^2(A)+\sin^2(B)-\sin^2(C)=2\sin(A)\sin(B) \cos(C)$ if $A+B+C=180$ degrees I most humbly beseech help for this question.
If $A+B+C=180$ degrees, then prove
$$
\sin^2(A)+\sin^2(B)-\sin^2(C)=2\sin(A)\sin(B) \cos(C)
$$
I am not sure what trig identity I should use to begin this problem.
| Let's take the Right Hand Side.
$$2 \sin B \cos C = \sin(B+C) + \sin(B-C) = \sin A + \sin(B-C)$$
Therefore, $2 \sin A \sin B \cos C = \sin^2 A + \sin A \sin(B-C)$
Now, $$2 \sin A \sin (B-C) = \cos (A - B +C) - \cos (A + B -C) = \cos 2C - \cos 2B = 2 \sin^2 B - 2 \sin ^2 C$$ Cancelling the $2$ gives us therefore, that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/177208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 1
} |
Which of the following statements are false Let$$A=\frac{1}{3}\begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix}$$
Which of the following statements are false?
(a) $A$ has only one real eigenvalue.
(b) $\operatorname{Rank}(A) = \operatorname{Trace}(A)$.
(c) Determinant of $A$ equals the determina... | You are totally right, except the third part. You do not have $A^2=A$, but instead you have that $det(A^n)=(det(A))^n=0^n=0=det(A)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/178268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$? Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality
$$\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$$?
| Although the function $f(x)=\sqrt{x}/(1-x)$ is not convex on (0,1), its tangent at $x=1/3$ lower bounds the
function and passes through the origin.
That is, for $0\leq x\leq 1$, we have
$${\sqrt{x}\over 1-x}\geq {3\sqrt{3}\over2}\, x.$$
Plugging in $a,b,c$ and adding gives
$${\sqrt{a}\over 1-a}+{\sqrt{b}\over 1-b}+{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/180937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
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Inequality $a+b+c \geqslant abc +2$ Assuming $a,b,c \in (0, \infty)$, we need to prove that:
$$a+b+c \geqslant a b c+2 \quad \text{if} \quad ab+bc+ca=3$$
Can you give me an idea, please? This inequality seem to be known, but I didn't manage to solve it.
| For $x,y,z \geq 0 $, $ f(x,y,z) = x+y+z $ given that, $xy+yz+xz =3 = \phi(x, y, z)$
$$ \nabla f = \lambda \nabla \phi $$
$$ 1 = \lambda (y+z) \hspace {2 cm} (1) $$
$$ 1 = \lambda (x+z) \hspace {2 cm} (2) $$
$$ 1 = \lambda (x+y) \hspace {2 cm} (3) $$
$$ xy+yz+xz =3 \hspace{2 cm} (4)$$
Solveing $(1), (2), (3), \text{ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 0
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How to evaluate the integral $\iint_C \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$ I would like to evaluate
$$\iint\limits_C \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$$
where $C$ is the first quadrant, i.e.
$$\int\limits_0^\infty\int\limits_0^\infty \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$$
where
$$\int\limits_0^\infty \sin x... | $$\begin{array}{c l}\int_0^\infty\int_0^\infty \sin(x^2+y^2)dxdy & = \int_0^\infty\int_0^\infty \sin(x^2)\cos(y^2)+\cos(x^2)\sin(y^2)dxdy \\[10pt] & =\int_0^\infty\sin(x^2)dx\int_0^\infty\cos(y^2)dy \\ & \qquad~~~+\int_0^\infty\cos(x^2)dx\int_0^\infty\sin(y^2)dy \\[10pt] & = 2\left(\frac{1}{2}\sqrt{\frac{\pi}{2}}\right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/181910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Getting square root of negative in completing the square problem I try to solve the equation $f(x) = 7x - 11 - 2x^2 = 0$ for $x$, but run into troubles. I've gone through it over and over again as well as similar problems, but can't find what I'm doing wrong.
$$f(x) = 7x - 11 - 2x^2 = 0$$
$$\iff x^2 - \frac{7}{2}x + \f... | You did everything fine but your quadratic equation has no real solutions, which you could have found out way more easily had you first calculated the equation's discriminant:
$$\Delta:=b^2-4ac=7^2-4(-2)(-11)=49-88=-39<0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/183731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to combine ratios? If $a:b$ is $2:5$, and $c:d$ is $5:2$, and $d:b$ is $3:2$, what is the ratio $a:c$? How would I go about solving this math problem?
if the ratio of $a:b$ is $2:5$ the ratio of $c:d$ is $5:2$ and the ratio of $d:b$ is $3:2$, what is the ratio of $a:c$?
I got $\frac{a}{c} = \frac{2}{5}$ but that is... | These ratios are just simple equations. For example $a:b=2:5$ is
$$a= \frac{2}{5}b$$
No need for confusing tricks here. Just substitutions :
$$ a = \frac{2}{5}b = \frac{2}{5}\frac{2}{3} d = \frac{2}{5}\frac{2}{3}\frac{2}{5} c = \frac{8}{75} c$$
So that
$$ a:c = 8:75 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/184669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Proof of $(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)>\frac{(ab+bc+ca)^3}{3}$ For positive real numbers $a$, $b$ and $c$, how do we prove that:
$$(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)>\frac{(ab+bc+ca)^3}{3}$$
| Just by sum of square and Schur's inequality:
$$
3(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2)-(ab+bc+ca)^3\\
=
\sum_{cyc} (a^{3}b+a^{3}c+a^{2}bc)(b-c)^2+3abc\sum_{cyc}a(a-b)(a-c) \geq 0
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/186575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
how can one solve for $x$, $x =\sqrt{2+\sqrt{2+\sqrt{2\cdots }}}$
Possible Duplicate:
Limit of the nested radical $\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$
how can one solve for $x$, $x =\sqrt[]{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2\cdots }}}}}}$
we know, if $x=\sqrt[]{2+\sqrt{2}}$, then, $x^2=2+\sqrt{2}$
now, if $x=\... | Define a sequence $\{a_n\}_{n \geq 1}$ such that $a_1 = \sqrt 2$ and $a_{n+1} = \sqrt{2 + a_n}$. It should be clear that $x$ is the limit of this sequence as $n$ goes to infinity, i.e:
$$x = \lim_{n \rightarrow \infty} a_n $$
To prove that this limit exists, it is sufficient to show that the sequence is bounded above, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/186652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 3
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Equation of line and its points
In the xy coordinate system if (a,b) and (a+3,b+k) are two points on the line defined by equation (the equation is kind of faded in text , but it seems to be like x=3y-7) then k =
A)9 , B)3 , C)1 , D)1 (Ans is 1)
Any suggestions on how that answer was calculated ?
| The slope of the line through $(a,b)$ and $(a+3, b+k)$ is $\frac{b+k-b}{a+3-a}$, which is $\frac{k}{3}$.
The slope of the line $x=3y-7$ is $\frac{1}{3}$. This is because the equation can be rewritten as $3y=x+7$, and then in standard slope-intercept form as $y=\frac{1}{3}x+\frac{7}{3}$.
These slopes are equal $\frac{k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/187762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does $i^i$ and $i^{1\over e}$ have more than one root in $[0, 2 \pi]$ How to find all roots if power contains imaginary or irrational power of complex number? How do I find all roots of the following complex numbers?
$$(1 + i)^i, (1 + i)^e, (1 + i)^{ i\over e}$$
EDIT:: Find the value of $i^i$ and $i^{1\over e}$ in $[0,... | Let $e^z=N≠0$ ,So, $N=e^z=e^z\cdot e^{2n\pi i}$ as $e^{2n\pi i}=1$ for any integer $n$.
$N=e^{z+2n\pi i}\implies Log N=z+2n\pi i$
If $n=0$ ,we get the principal value $z=\log N$, so $LogN=\log N+2n\pi i$
Let $A=(a+ib)^{x+iy}$
$Log A=(x+iy)Log(a+ib)$
Let $a=r\cos \theta$ and $b=r\sin \theta\implies a+ib=r(\cos \theta+i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/189703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find the angle between two bivariate functions.. Find the angle between $x^2+y^2=8$ and $xy=4$ at the intersection points.
I have thought in find the angle between the tangent lines of each function at the intersection point, but i don´t know how to do it
| Let the intersection point be $(a,b)$, so $a^2+b^2=8$ and $ab=4≠0$
$$\implies \frac{a^2+b^2}{ab}=\frac{4}{2}\implies (a-b)^2=0\implies a=b≠0$$
So,$a=±2$ as $ab=4$.
$$x^2+y^2=4\implies 2x+2y\frac{dy}{dx}=0\implies (\frac{dy}{dx})_{x=y}=-1$$
$$xy=4\implies x\frac{dy}{dx}+y=0\implies (\frac{dy}{dx})_{x=y}=-1 $$
So, at $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/192282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $ 1 + 2 + 4 + 8 + \dots = -1$
Possible Duplicate:
Infinity = -1 paradox
I was told by a friend that $1 + 2 + 4 + 8 + \dots$ equaled negative one. When I asked for an explanation, he said:
Do I have to?
Okay so, Let $x = 1+2+4+8+\dots$
$2x-x=x$
$2(1+2+4+8+\dots) - (1+2+4+8+\dots) = (1+2+4+8+\dots)$
Therefore, ... | Look at Calculus textbook, undergraduate level. When we treat infinite sum, we cannot change the order to compute.
Example. $1-1+1-\cdots$
$$(1-1)+(1-1)+\cdots=0+0+\cdots=0$$
$$1+(-1+1)+(-1+1)+\cdots=1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/193872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 0
} |
Calculation of a strange series Is it possible to find an expression for:
$$S(N)=\sum_{k=0}^{+\infty}\frac{1}{\sum_{n=0}^{N}k^n}?$$
For $N=1$ we have
$$S(1) = \displaystyle\sum_{k=0}^{+\infty}\frac{1}{1 + k} = \displaystyle\sum_{k=1}^{+\infty}\frac{1}{k}$$
which is the (divergent) harmonic series. Thus, $S (1) = \infty... | Let $T(N) = S(N-1)$. Then
$$ \begin{align*}T(n)
&= 1 + \frac{1}{n} + \sum_{k=2}^{\infty} \frac{1}{k^{n-1}+k^{n-2}+\cdots+k+1} \\
&= 1 + \frac{1}{n} + \sum_{k=2}^{\infty} \frac{k - 1}{k^n - 1} \\
&= 1 + \frac{1}{n} + \sum_{k=2}^{\infty} \frac{1}{n} \sum_{l=1}^{n-1} \frac{\omega_l (\omega_l - 1)}{k - \omega_l} \\
&= 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/194096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 1
} |
Estimation of a polynomial I'm currently reading the following paper: http://arxiv.org/abs/1209.0612 and got stuck on Proposition 3.1 (2).
The claim translated to polynomials is the following:
Assume $n\geq 3, c\geq 1, d\geq 1$ are natural numbers such that $c²+d²-(n-1)cd<0$. Show that $(n³-n+1)c²+(n+1)d²-(n²+n-1)cd>... | Let $c,d$ be positive real numbers and $n>1$ (esp., $n^3-n+1>0$).
By the arithmetic-geometric inequality
$$(n^3-n+1)c^2+(n+1)d^2\ge 2\cdot\sqrt{(n^3-n+1)(n+1)}\cdot c d.$$
One checks by multiplying out that
$$4(n^3-n+1)(n+1)=(n^2+n-1)^2+3+3n^2(n^2-1)+2n(n^2+1),$$
hence $2\cdot\sqrt{(n^3-n+1)(n+1)}>n^2+n-1$ and finally... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/195261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to prove this inequality $\sqrt{\frac{ab+bc+cd+da+ac+bd}{6}}\geq \sqrt[3]{{\frac{abc+bcd+cda+dab}{4}}}$ How to prove this inequality
$$\sqrt{\frac{ab+bc+cd+da+ac+bd}{6}}\geq \sqrt[3]{{\frac{abc+bcd+cda+dab}{4}}} $$
for $a,b,c,d\gt0$?
Thanks
| Here is my proof for positive variables from 1979.
Let $a+b+c+d=4u$, $ab+ac+bc+ad+bd+cd=6v^2$, $abc+abd+acd+bcd=4w^3$ and $abcd=t^4$, where $v>0$.
Hence, $a$, $b$, $c$ and $d$ are positive roots of the following equation:
$(x-a)(x-b)(x-c)(x-d)=0$ or $x^4-4ux^3+6v^2x^2-4w^3x+t^4=0$.
Hence, by Rolle the equation $(x^4-4u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/197955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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$(1+i)$ to the power $n$
Possible Duplicate:
Complex number: calculate $(1 + i)^n$.
I came across a difficult problem which I would like to ask you about:
Compute
$ (1+i)^n $ for $ n \in \mathbb{Z}$
My ideas so far were to write out what this expression gives for $n=1,2,\ldots,8$, but I see no pattern such that I ca... | You have a closed form already if you do just a little more work:
$$\begin{align*}
(1+i)^n&=\left(\sqrt{2}\right)^n\left(\cos\frac{n\pi}4+i\sin\frac{n\pi}4\right)\\
&=2^{n/2}\left(\cos\frac{n\pi}4+i\sin\frac{n\pi}4\right)\\
&=\begin{cases}
2^{n/2},&\text{if }n\equiv 0\pmod 8\\
2^{n/2}\left(\frac{\sqrt2}2+\frac{\sqrt2}2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/199004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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the least possible value for :$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $ If we know that for every $a,b,c>0$ ,how we can find the least possible value for :
$$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $$
| The expression
$$
E := \frac {a + b} c + \frac {b + c} a + \frac {c + a} b
$$
is homogeneous, so we can assume $a + b + c = 1$. Then it becomes
$$
E = \frac 1 a + \frac 1 b + \frac 1 c - 3
$$
By CS inequality we get
$$
\frac 1 a + \frac 1 b + \frac 1 c \geq \frac {(1 + 1 + 1)^2} {a + b + c} = 9
$$
So $E \geq 6$. By th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/200289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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A inequality proposed at Zhautykov Olympiad 2008 An inequality proposed at Zhautykov Olympiad 2008.
Let be $a,b,c >0$ with $abc=1$. Prove that:
$$\sum_{\mathrm{cyc}}{\frac{1}{(a+b)b}} \geq \frac{3}{2}.$$
Set $a=\frac{x}{y}$, $b=\frac{y}{z}$, $c=\frac{z}{x}$.
Our inequality becomes:
$$\sum_{\mathrm{cyc}}{\frac{z^2... | Let $a=\frac{x}{y}$ and $b=\frac{y}{z}$, where $x$, $y$ and $z$ be positives.
Hence, since $abc=1$, we get $c=\frac{z}{x}$ and by C-S we obtain:
$$\sum_{cyc}\frac{1}{(a+b)b}=\sum_{cyc}\frac{z^2}{xz+y^2}=\sum_{cyc}\frac{z^4}{xz^3+z^2y^2}\geq\frac{(x^2+y^2+z^2)^2}{\sum\limits_{cyc}(x^2y^2+x^3y)}.$$
Thus, it remains to pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/202053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Help me proof the identity $\sin^2\alpha+\cos^2\alpha=1$ Prove the identity $\sin^2\alpha+\cos^2\alpha=1$. Thanks
| $$\sin\alpha=\frac{a}{c}\Rightarrow\sin^2 \alpha=\frac{a^2}{c^2}$$
$$\cos\alpha=\frac{b}{c}\Rightarrow\cos^2\alpha=\frac{b^2}{c^2}$$
$$\sin^2\alpha+\cos^2\alpha=\frac{a^2}{c^2}+\frac{b^2}{c^2}=\frac{a^2+b^2}{c^2}=\frac{c^2}{c^2}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/203473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Ideas on the ways to integrate $\int \tan^2( x)\sec^3( x) dx$ I would proceed by thus , let $y = [\sec (x)]^2 $
then
$$dy = 2 \cdot \sec(x) \cdot \sec(x) \cdot \tan(x) \cdot dx = 2 \cdot ( \sec (x))^2 \cdot \tan(x) \cdot dx $$
so,
$$
2 \tan^2(x) \sec^2 (x) dx = \sec(x) \cdot \tan(x) \cdot dy = y(y-1)^\frac{1}{2} \cdo... | Let you want to solve $\int R\big(\sin(x),\cos(x)\big)dx$ and you know that $$R\big(\sin(x),-\cos(x)\big)\equiv -R\big(\sin(x),\cos(x)\big)$$ then you can take $\sin(x)=t$ for a good substitution. We have here $$\int \tan^2(x)\sec^3(x)dx=\int \frac{\sin^2(x)dx}{\cos^5(x)}$$ and we can see the above statement is true fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/203861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Vector Project onto Subspace So the question is:
Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} \frac{1}{\sqrt{2}}\\\frac{-1}{\sqrt{2}}\\0\end{pmatrix}$. Let $x=(1,2,2)^T$. Find the projection p of x onto S.... | Let $\{v_1,\dotsc,v_m\}$ be a basis for a subspace $V$ of $\Bbb R^n$. The matrix of the orthogonal projection onto $V$ is
$$
P=A\left(A^\top A\right)^{-1}A^\top
$$
where $A$ is the matrix whose columns are $\{v_1,\dotsc,v_m\}$. A nice discussion of this can be found in these notes.
In our case we have
$$
A=
\left[\begi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Solve the dirichlet pde with the given conditions. Solve the Dirichelt problem:
$$\nabla^2u(x,y)=0$$
$$0\le x\le3,0<y<7$$
$$$u(x,0)=0, u(x,7)=sin((\pi)x/3$$ $$9\le x\le3$$
$$u(0,y)=u(3,y)=0$$
$$0\le y\le7$$
Using separation of variables I found that $X(0)=0$ and $X(3)=0$ because $u(0,y)=u(3,y)=0$. Also, $u(x,0)=0$ s... | Although your approach is not completely wrong, your approach is not the best.
Note that for $\nabla^2u(x,y)=0$ with conditions of the types $u(x,0)$ , $u(x,7)$ , $u(0,y)$ and $u(3,y)$ , according to http://eqworld.ipmnet.ru/en/solutions/lpde/lpde301.pdf#page=2 , we have special consideration:
$u(x,y)=\sum\limits_{n=1}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Elementary Question about limits Prove
$$
\lim_{x \to 0}\frac{x^3 - \sin^3x}{x - \ln{(1+x)} - 1 + \cos x} = 0
$$
Obviously, Using l'Hôpitals rule, we can evaluate this limit. But, taking derivatives of such functions is such a mess. Anyone sees a trick to do this problem faster? any ideas?
| We have $x^3-\sin^3x=(x-\sin x)(x^2+x\sin x+\sin^2x)\approx\frac{x^3}{6}(3x^2)=\displaystyle\frac{x^5}{2}$ as $x\to 0$.
Hence
\begin{equation*}
\begin{array}{lll}
\displaystyle\lim_{x \to 0}\frac{x^3 - \sin^3x}{x - \ln{(1+x)} - 1 + \cos x} &=&\displaystyle\lim_{x\to 0}\frac{x^5}{2(x - \ln{(1+x)} - 1 + \cos x)}\\
&\over... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/214579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
proof by by directly manipulating the sum: Prove that
$$\sum_{i=0}^{n} x ^{i} = \frac{ 1-x ^{n+1} }{ 1-x }$$
to be used by directly manipulating the sum: let A be the sum, and show that xA = A + x^(n+1) -1
I don't get how its going to equal $\frac{ 1-x ^{n+1} }{ 1-x }$
$$xA=x\sum_{i=0}^n x^i=x(x^0+x^1+x^2+x^3+...)=x... | Write it out like this:
$$\begin{align*}
A&=x^0+\color{red}{x^1+x^2+\ldots+x^n}\\
xA&=\quad\quad\,\color{red}{x^1+x^2+\ldots+x^n}+x^{n+1}\\
A-xA&=x^0+\qquad\qquad\color{red}{0}\qquad\quad\,-x^{n+1}
\end{align*}$$
Then $(1-x)A=A-xA=x^1-x^{n+1}=1-x^{n+1}$, so $$A=\frac{1-x^{n+1}}{1-x}\;.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/215210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove the trigonometric identity $(35)$ Prove that
\begin{equation}
\prod_{k=1}^{\lfloor (n-1)/2 \rfloor}\tan \left(\frac{k \pi}{n}\right)= \left\{
\begin{aligned}
\sqrt{n} \space \space \text{for $n$ odd}\\
\\
\ 1 \space \space \text{for $n$ even}\\
\end{aligned}
\right.
\end{e... | $$\tan nx=\frac{^nC_1t-^nC_3t^3+^nC_5t^5-\cdots }{^nC_0t^0-^nC_2t^2+^nC_4t^4-\cdots }$$ where $t=\tan x$
If $\tan nx=0, x=\frac {k\pi}n$ where $0\le k< n$, clearly, the roots of this $n$-degree equation are $\tan\frac{k\pi}n$
If $n$ is odd,
$^nC_n(-1)^{\frac{n-1}2}t^n+^nC_{n-2}(-1)^{\frac{n-3}2}t^{n-2}+\cdots-^nC_3t^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/218766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 3
} |
Differentiating $x^2 \sqrt{2x+5}-6$ How do I differentiate this function: f(x)= $x^2 \sqrt{2x+5}-6$
I had: I had $2x\sqrt{2x+5} + x^2 \dfrac{1}{2\sqrt{2x+5}}$ but the correction model said it was I had $2x\sqrt{2x+5} + x^2 \dfrac{2}{2\sqrt{2x+5}}$
| I would use the product rule, which you seem to have tried from your comment above. Here would be the idea:
$\begin{split}
\frac{d[x^2 \sqrt{2x+5} - 6]}{dx}
&= \frac{dx^2}{dx} \sqrt{2x+5} + x^2 \frac{d(2x+5)^{\1/2}}{dx} \\
&= 2x \sqrt{2x+5} + x^2 \frac{1}{2} (2x+5)^{-1/2} \cdot 2 \\
&= 2x \sqrt{2x+5} + \frac{x^2}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/220858",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Evaluate the following integral, $\int\sqrt{4-\sqrt{x}}dx$ Evaluate the following integral,
$$\int\sqrt{4-\sqrt{x}}dx$$
$$\int \sqrt{4-\sqrt{x}}dx=\int \sqrt{2^2-(x^{1/4})^2}dx$$
Considering the common subsitution for $a^2-x^2$, let $$x^{1/4}=2\sin t$$
$$x=16\sin^4t$$$$\int dx=\int 64\sin^3t\cos t dt$$
Theref... | So, substitute $u = \sqrt{x}$ and $\mathrm{d}u = \frac{1}{2 \sqrt{x}} \,\mathrm{d}x$:
$$= 2 \int \!\sqrt{4-u}\, u \, \mathrm{d}u$$
For the integrand $\sqrt{4-u}\, u$, substitute $s = 4-u$ and $\mathrm{d}s = - \mathrm{d}u$:
$$= 2 \int \!(s-4) \sqrt{s}\, \mathrm{d}s$$
Expanding the integrand $(s-4) \sqrt{s}$ gives $s^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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How to find a parametric equation for the tangent line to the curve of intersection of the cylinders? How can i find a parametric equation for the tangent line to the curve of intersection of the cylinders $x^2 + y^2 = 4$ and and $x^2 + z^2 = 1$ at the point $P_0(1,\sqrt{3}, 0)$?
| The equation of any line passing through $(1,\sqrt 3,0)$ can be written as
$\frac{x-1}a=\frac{y-\sqrt 3}b=\frac z c$ where $a^2+b^2+c^2=1$
So,$cx=az+c,cy=bz+\sqrt 3c$
Putting the values of $x,y$ in $x^2+y^2=4,$
$(az+c)^2+(bz+\sqrt 3c)^2=4c^2$
$(a^2+b^2)z^2+2zc(a+\sqrt 3 b)=0$
But this is a quadratic in $z,$, each root... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/225398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
sum of a series Can
\begin{equation}
\sum_{k\geq 0}\frac{\left( -1\right) ^{k}\left( 2k+1\right) }{\left(
2k+1\right) ^{2}+a^{2}},
\end{equation}
be summed explicitly, where $a$ is a constant real number? If $a=0,$ this
sum becomes
\begin{equation}
\sum_{k\geq 0}\frac{\left( -1\right) ^{k}}{2k+1}=\frac{\pi }{4}.
\end{e... | The series is summable and has the closed form formula
$$-\frac{1}{4}\,{\frac {\pi }{\sin \left( \frac{1}{2}\pi \, \left( 3 + ia \right )
\right) }} = \frac{1}{4}{\frac {\pi }{\cosh \left( \frac{\pi \,a}{2} \right) }} .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/226308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
Definite integral involving Fresnel integrals I am seeking to evaluate
$\int_0^{\infty} f(x)/x^2 \, dx$
with
$f(x)=1-\sqrt{\pi/6} \left(\cos (x) C\left(\sqrt{\frac{6 x}{\pi }} \right)+S\left(\sqrt{\frac{6 x}{\pi }} \right) \sin
(x)\right)/\sqrt{x}$.
$C(x)$ and $S(x)$ are the Fresnel integrals. Numerical integration ... | First of all,
$$
C(x)=\int_{0}^{x}\cos\left(\frac{1}{2}\pi t^{2}\right)dt=\sqrt{\frac{2}{\pi}}\int_{0}^{\sqrt{\pi/2} \,\left (x\right )}\cos(z^{2})\, dz,
$$
and
$$
S(x)=\int_{0}^{x}\sin\left(\frac{1}{2}\pi t^{2}\right)dt=\sqrt{\frac{2}{\pi}}\int_{0}^{\sqrt{\pi/2} \,\left (x\right )}\sin(z^{2})\, dz.
$$
So we have
$$
C\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/227641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Prove the following relation: I must prove the relation $$\sum_{k=0}^{n+1}\binom{n+k+1}{k}\frac1{2^k}=2\sum_{k=0}^n\binom{n+k}{k}\frac1{2^k}.$$
I got this far before I got stuck:
$\begin{eqnarray*}
\sum_{k=0}^{n+1}\binom{n+k+1}{k}\frac1{2^k} & = & \sum_{k=0}^{n+1}\left\{\binom{n+k}{k}+\binom{n+k}{k-1}\right\}\frac1{2^k... | Let $s_n=\displaystyle\sum_{k=0}^n\binom{n+k}k\frac1{2^k}$. Then
$$\begin{align*}
s_{n+1}&=\sum_{k=0}^{n+1}\binom{n+k+1}k\frac1{2^k}\\
&=\sum_{k=0}^{n+1}\left(\binom{n+k}k+\binom{n+k}{k-1}\right)\frac1{2^k}\\
&=\binom{2n+1}{n+1}\frac1{2^{n+1}}+\sum_{k=0}^n\binom{n+k}k\frac1{2^k}+\sum_{k=0}^n\binom{n+1+k}k\frac1{2^{k+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/228339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Permutation and cycles I need to determine the missing number to fulfill the following reproduction:
$$\pi=\pmatrix{1&2&3&4&5&6&7&8&9\\3&5&9&4&1&2&6&7&8}$$
And here is the representation of $pi$ as the product of transpositions:
$$(1 i) (1 3) (2 5) (9 8) (8 7) (7 6) (3 6) = \pi$$
The result for $i$ must be 2 but I have... | According to the comments, you’re multiplying from right to left. If you multiply the transpositions that you know completely, you get
$$\pmatrix{6&3&7&8&9&5&2&1&4\\1&9&6&7&8&2&5&3&4}\;,$$
which we rearrange into standard form as
$$\pmatrix{1&2&3&4&5&6&7&8&9\\3&5&9&4&2&1&6&7&8}\;.\tag{1}$$
We want to choose $i$ so that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/229015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Probability that I will get a $4$ consecutive numbers I have five dice to roll. I roll them. What is the probability that I will get a straight with exactly four consecutive numbers and not $5$?
There are three options: $1,2,3,4$ or $2,3,4,5$ or $3,4,5,6$.
I have $1,2,3,4,*$ where $*$ can be either $1/2/3/4/6$. It cann... | Imagine tossing the dice one at a time, and recording the results, or equivalently labelling the dice A to E, and recording the results as a string of length $5$, result on A, result on B, and so on. There are $6^5$ possibilities, all equally likely. Now we count the favourables.
The "straight" part can be of any of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/231122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Induction: $\sum_{k=0}^n \binom nk k^2 = n(1+n)2^{n-2}$ I found crazy (for me at least) induction example, in fact it just would be nice to prove. (Even have problems with starting) Any hints are highly valued: $$0^2\binom{n}{0}+1^2\binom{n}{1}+2^2\binom{n}{2}+\cdots+n^2\binom{n}{n}=n(1+n)2^{n-2} $$
| Marvis has given a typically excellent answer. I'll go ahead and show you an induction-style proof, just in case you're interested.
A useful basic combinatoric fact for this induction proof is Pascal's identity: $$\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}\tag{1}$$ Another nice basic fact is $$\sum_{k=0}^n\binom{n}k=2^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/231596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 7,
"answer_id": 4
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Find $2\times 2$ matrix of linear transformation $T$ Given:
\[
T\left(\begin{bmatrix}-1 \\ -2\end{bmatrix}\right) = \begin{bmatrix}17 \\ 11\end{bmatrix} \text{ and } T\left(\begin{bmatrix}2 \\ 3\end{bmatrix}\right) = \begin{bmatrix}-30 \\ -17\end{bmatrix}
\]
Find a matrix such that:
\[ T(v) = \begin{bmatrix}? & ? \\ ... | Let's write $$\begin{bmatrix} a & b \\ c & d\end{bmatrix}$$ for the matrix we want to find. Then the two given equations read
\begin{align*}
-a - 2b &= 17\\
-c - 2d &= 11\\[3mm]
2a + 3b &= -30\\
2c + 3d &= -17
\end{align*}
We have to solve this system, so let's first look at the equations for $a$ and $b$
\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/233412",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Finding all real zeros of the polynomial $$x^5 - 5x^4 +6x^3 -30x^2 +8x - 40 = 0$$
So far I have...
$$r/s: +- 1, +- 40, +- 2, +- 20, +- 4, +- 10, +-5, +- 8$$
Only $+ 5$ works.
Then I have $$(x + 5)( ) = x^5 - 5x^4 +6x^3 -30x^2 +8x - 40$$
Then you have to use long devision between $x + 5$ and $ x^5 - 5x^4 +6x^... | Hint:
$$x^5-5x^4+6x^3-30x^2+8x-40=x^4(x-5)+6x^2(x-5)+8(x-5)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/237389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $a,b,c,d$ be the roots of the biquadratic $x^4-x^3+2x^2+x+1=0$ then show that $(a^3+1)(b^3+1)(c^3+1) (d^3+1)=16$ If $a,b,c,d$ be the roots of the biquadratic $x^4-x^3+2x^2+x+1=0$ then show that $(a^3+1)(b^3+1)(c^3+1) (d^3+1)=16$
I have tried to solve the equation first and find the values of the roots but it become... | $$\begin{align}
a+b+c+d &= \phantom{-}1 \\
ab+bc+cd+ac+ad+bd &= \phantom{-}2 \\
abc+bcd+abd+acd &= -1 \\
abcd &= -1
\end{align}$$
All you need to find is
$$\begin{align}
(abc)^3 + (abd)^3 + (acd)^3 + (bcd)^3 &= p \\
(ab)^3 + (ac)^3 + (ad)^3 + (bc)^3 + (bd)^3 + (cd)^3 & = q \\
a^3 + b^3 + c^3 +d^3 &= r
\end{align}$$
And... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/241751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Why is the determinant of a symplectic matrix 1? Suppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix}
0 & E_n\\
-E_n&0
\end{pmatrix}$$
where $E_n$ represents identity matrix.
if $A$ satisfies $$AJA^T=J.$$
How to figure out $$\det(A)=1~?$$
My approach:
I have tried to separate $A$ into four submartix:$$A=\begin... | There is an easy proof for real and complex case which does not require the use of Pfaffians. This proof first appeared in a Chinese text. Please see http://arxiv.org/abs/1505.04240 for the reference.
I reproduce the proof for the real case here. The approach extends to complex symplectic matrices.
Taking the determin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/242091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 7,
"answer_id": 0
} |
Solve for x $\left\lfloor x \right\rfloor^3+2x^2=x^3+2\left\lfloor x\right\rfloor^2$ Solve for x
$$\left\lfloor x \right\rfloor^3+2x^2=x^3+2\left\lfloor x\right\rfloor^2$$
where $\left\lfloor t \right\rfloor$ denotes the largest integer not exceeding t
$X \in \mathbb{Z}$ is a solution. Is there other root? Thanks.
| Write the equation as $x^3 - 2x^2 = \lfloor x \rfloor^3 - 2\lfloor x \rfloor^2$. Clearly this has solutions in $\mathbb{Z}$.
On most intervals of the form $[n, n+1]$ with $n \in \mathbb{Z}$, the function $x^3 - 2x^2$ is monotone. You will only get interesting values on the intervals where it isn't.
To find those, take ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Minimum Possible Number How to find the minimum possible number of length N ,which is simultaneously divisible by the single digit prime number like 2,3,5,7 ?like of length 5 minimum possible number is 10080.
| As the lcm of $2,3,5,7$ is $210,$ we need to find the minimum multiple in $N$ digits.
So, $N$ must be $\ge 3$ to admit solution.
The minimum natural number with $N$ digits is $M=100\cdots00$ with $(N-1)$ zeros.
So, if $D=\lfloor\frac M {210}\rfloor,$ the answer will be $210(D+1)$
For example if $N=4,D=\lfloor\frac {100... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/244345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Dual of a Linear Program \begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b
\end{align}
Note that here $x$ is unrestricted. I need to prove that the dual of this program is given by
\begin{align}
\max_{\lambda} \lambda^Tb \\
s.t.~\lambda^TA\leq c^T
\end{align}
But in the constraint, I always get an equality (using what I learnt)... | \begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b \\
Unrestricted
\end{align}
Take $x=x_1-x_2$
\begin{align}
\min c^T(x_1-x_2) \\
s.t.~A(x_1-x_2)=b \\
~ x_1, x_2 \ge 0
\end{align}
This is can be written as
\begin{align}
\min {\begin {pmatrix}c \\ -c \\ \end {pmatrix} }^T \begin {pmatrix} x_1 \\ x_2 \\ \end {pmatrix} \\
s.t.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/244875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Proving $\lim\limits_{x\to a}(a-x)\int\limits_0^x\frac{f(y)}{(a-y)^2}\,dy=f(a)$ How to prove for a continuous function $f$, the following limit holds?
$$\lim_{x\to a}\,(a-x)\int_0^x\frac{f(y)}{(a-y)^2}\,dy=f(a)$$
| Here is general proof without using limit theorems:
First remark that
$$(x-a) \cdot \int_0^x \frac{f(a)}{(a-y)^2} = f(a)+ \frac{f(a) \cdot (x-a)}{a}$$
hence
$$\left|(a-x) \cdot \int_0^x \frac{f(y)}{(a-y)^2} \, dy - f(a) \right| \leq \left| (a-x) \cdot \int_0^x \frac{f(y)}{(a-y)^2} \, dy - f(a) + \frac{f(a) \cdot (x-a)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/246401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
What am I doing wrong in calculating this determinant? I have matrix:
$$
A = \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix}
$$
And I want to calculate $\det{A}$, so I have written:
$$
\begin{array}{|cccc|ccc}
1 & 2 & 3 & 4 & 1 & 2 & 3 \\
2 & 3 & 3 & 3 & 2 & 3 & 3 \\
0 & ... | The others have pointed out what's wrong with your solution. Let's calculate the determinant now:
\begin{align*}
\det \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix} &\stackrel{r1 - \frac12(r2+r3+r4)}{=}
\det \begin{bmatrix}
0 & 0 & 0 & 0 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/246606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Compute the limit Consider an equation
$$
\tan (x) = \frac{a x}{x^2+b}
$$
where $a,b \neq 0$. Plotting $\tan(x)$ and function on the RHS we can see that this equation has infinitely many positive solutions $x_{n}$ and $x_{n} \sim \pi n$ as $n \to \infty$, i.e.
$$
\lim\limits_{n \to \infty} \frac{x_{n}}{n} = \pi.... | Let us assume $a,b>0$. The proof can be easily adapted if $a,b$ are not positive.
Consider $f(x) = \dfrac{ax}{x^2+b} - \tan(x)$. Note that $f(x)$ is nice except at $x = m \pi + \pi/2$ i.e. $$f((m \pi + \pi/2)^-) = - \infty \,\,\,\,\,\, f((m \pi + \pi/2)^+) = \infty$$
$$f'(x) = \dfrac{a(b-x^2)}{(b+x^2)^2} - \sec^2(x)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/246802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Limit $\lim_{n \rightarrow \infty}\frac{1 \cdot 3 \cdot 5 \dots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \dots \cdot (2n)} $ $\displaystyle \lim_{n \rightarrow \infty}\frac{1 \cdot 3 \cdot 5 \dots \cdot (2n-1)}{2 \cdot 4 \cdot 6 \cdot \dots \cdot (2n)} $
| $$\dfrac12 \cdot \dfrac34 \cdot \dfrac78 \cdots \dfrac{2n-1}{2n} = \left(1 - \dfrac12\right)\left(1 - \dfrac14\right)\left(1 - \dfrac16\right)\left(1 - \dfrac18\right)\cdots\left(1 - \dfrac1{2n}\right)$$
Since $$\dfrac12 + \dfrac14 + \dfrac16 + \cdots + \dfrac1{2n} + \cdots$$ diverges, the infinite product goes to $0$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/247360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Differential equation problems I need your help to finish my homework. Can somebody help me?
I cannot finish all my homework especially in this problem.
*
*$\dfrac{dy}{dx} = \dfrac{y^2 -1}{x}$; my answer is $-\dfrac{1}{y} -y = \ln x + x$, is it right?
*$\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$
*$\dfrac{dy}{dx} = \... | For 2. The equation can be formed as:$$\frac{dy}{dx}=\frac{x^2\left(1+\frac{y^2}{x^2}\right)}{x^2\left(\frac{y}{x}\right)}$$
then if $x\neq0$ by taking $u=\frac{y}{x}$, we have $$\frac{dy}{dx}=\frac{\left(1+u^2\right)}{u}$$
but $u=\frac{y}{x}$ leads us to $xu=y$ and then $1+u'=y'$ where in $y'=\frac{dy}{dx}$. Now we ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/251600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"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.