Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Show that $(x + 1)^{(2n + 1)} + x^{(n + 2)}$ can be divided by $x^2 + x + 1$ without remainder I am in my pre-academic year. We recently studied the Remainder sentence (at least that's what I think it translates) which states that any polynomial can be written as $P = Q\cdot L + R$
I am unable to solve the following:
... | $$(x+1)^{2n+1}+x^{n+2}=(x+1)\{(x+1)^2\}^n+x^{n+2}=(x+1)(x^2+2x+1)^n+x^{n+2}$$
Now as $x^2+2x+1\equiv x\pmod{x^2+x+1}$
$$(x+1)(x^2+2x+1)^n+x^{n+2}\equiv (x+1)(x)^n+x^{n+2}\pmod{x^2+x+1}$$
$$\equiv x^n(x+1+x^2)$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Quadratic equations and inequalities $\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$ and $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$ For every positive integer $n$, prove that
$$\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$$
Hence or otherwise, prove that $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$, where $[x]... | $$\sqrt n+\sqrt{n+1}>\sqrt{4n+1}\\n+2\sqrt{n(n+1)}+n+1>4n+1\\\sqrt{n(n+1)}>n\\n^2+n>n^2\\n>0\\\sqrt{4n+2}>\sqrt n+\sqrt{n+1}\\4n+2>2n+1+2\sqrt{n(n+1)}\\2n+1>2\sqrt{n(n+1)}\\4n^2+4n+1>4n^2+4n\\1>0$$
| {
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When is the sum of consecutive squares a prime? For what integers $x$ do there exist $x$ consecutives integers, the sum of whose squares is prime?
I tried use $$1^2+2^2+...+n^2=\frac {n(n+1)(2n+1)}{6}$$
| Let our numbers be $a^2,(a+1)^2,\dots,(a+x-1)^2$. The sum of the squares is $a^2 x+2a(1+\cdots+(x-1))+(1^2+\cdots+(x-1)^2)$.
Note that $a^2 x$ is divisible by $x$, as is $2a(1+\cdots+(x-1))$. So we concentrate on the term $1^2+2^2+\cdots+x^2$. Call this number $N$. By the formula quoted in the post, we have
$$6N=(x-1)... | {
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Associated Legendre functions special values I should prove that
$$P_n^n(\cos \theta)=(2n-1)!! \sin^n\theta$$
$$P_n^m(0)=\begin{Bmatrix} (-1)^{(m+n)/2}\displaystyle\frac{(n+m-1)!!}{(n-m)!!} & \mbox{ if }& n+m \text{ even}\\ 0 & \mbox{if}& n+m \text{ odd}\end{Bmatrix}$$
$P_n^m(x)$ is a associated Legendre functions
I do... | The associated Legendre polynomials $P_l^m$ (which are actually not polynomials for odd m :-)) are given by
$$P_l^m(x) = \frac{(-1)^m}{2^ll!}(1-x^2)^{m/2}[(x^2-1)^l]^{(l+m)}$$
Where $f^{(n)}$ stands for the $n^{th}$ derivative of $f$. In this formula, $x = cos \theta \in[-1,1]$. We derive
\begin{equation}
\begin{split}... | {
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Odd series convergence Prove that we have following inequality:
$1+ \frac{1}{3} + \frac{1}{5} + ... + \frac{1}{397} > \frac{9}{4}$
Anybody can help me to figure it out?
| A silly approach:
$$
\sum_{n=0}^{198} \frac{1}{2n+1} > \int_0^{198} \frac{dx}{2x+1} = \frac{1}{2} \log 397 > \frac{1}{2} \log 361 = \log 19,
$$
$$
e^{9/4} < 3^{9/4} < 3^{10/4} = 9 \sqrt{3} < 9 \cdot 2 = 18.
$$
A more sensible approach:
Write
$$
\sum_{n=1}^{199} \frac{1}{2n-1} > \sum_{n=1}^{\large 2^7} \frac{1}{2n-1} > ... | {
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Solving this linear second order Cauchy problem Here is my problem, I know that a function is a solution of this linear Cauchy problem
$$
\left\{
\begin{array}{rcl}
y'' &=& \frac{x^2+6}{4}y,\\
y(0)&=&0,\\
y'(0)&=& \frac{\sqrt{\pi}}{2}.
\end{array}
\right.
$$
and I want to find the solution of this problem with a simpl... | Apply the method in http://eqworld.ipmnet.ru/en/solutions/ode/ode0205.pdf:
Let $y=e^\frac{x^2}{4}~u$ ,
Then $y'=e^\frac{x^2}{4}~u'+\dfrac{xe^\frac{x^2}{4}}{2}u$
$y''=e^\frac{x^2}{4}~u''+\dfrac{xe^\frac{x^2}{4}}{2}u'+\dfrac{xe^\frac{x^2}{4}}{2}u'+\biggl(\dfrac{x^2e^\frac{x^2}{4}}{4}+\dfrac{e^\frac{x^2}{4}}{2}\biggr)u=e^... | {
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Prove the inequality $\frac{a^2+1}{b+c}+\frac{b^2+1}{a+c}+\frac{c^2+1}{a+b}\ge 3$
If $a,b,c\in\mathbb R^+$ prove that:
$$\frac{a^2+1}{b+c}+\frac{b^2+1}{a+c}+\frac{c^2+1}{a+b}\ge 3$$
| Apply Am-GM to the numerator of each fraction.
You get the statement of Nesbit inequality.
| {
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Sturm Liouville with periodic boundary conditions Background and motivation: I'm given the boundary value problem:
$$y''(x)+2y(x)=-f(x)$$
subject $y(0)=y(2\pi)$ and $y \, '(0)=y \, '(2\pi)$.
EDIT: These were not given to be zero !! Maybe this helps...
The text (Nagle Saff and Snider, end of Chapter 11 technical writ... | The Green's function is the solution when $f(x)=\delta(x-x_s)$, where $x_s$ is some kind of point source position that forces the system. Let's suppose that $x_s\in(0,2\pi)$. For $x\neq x_s$, the delta function is zero, and so we solve the homogeneous equation
$$
\left|
\begin{array}{cc}
y'' + 2y = 0, & x<x_s\\
y'' + 2... | {
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Computing the Gaussian curvature of this surface $z=e^{(-1/2)(x^2+y^2)}$. Compute the Gaussian curvature of $z=e^{(-1/2)(x^2+y^2)}$. Sketch this surface and show where $K=0 $, $K>0$, and $K<0$.
So would the easiest way to do this question be to construct a parametrization $$\mathbf{x}(u,v)=(u, v, e^{-\frac{1}{2}(u^2+v^... | The parametrization $$\sigma = (u,v, e^{-(x^2+y^2)/2})$$ would indeed be the simplest parametrization, but not for computational purposes.
So we can also let $x = r\cos\theta$ and $y = r\sin\theta$ and get,
$$\sigma = (r\cos\theta,r\sin\theta, e^{-r^2/2}).$$
After finding the normal vector, we get
$$\mathbf{N} = \frac{... | {
"language": "en",
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Integral: $\int \frac{dx}{\sqrt{x^{2}-x+1}}$ How do I integrate this? $$\int \frac{dx}{\sqrt{x^{2}-x+1}}$$
I tried solving it, and I came up with $\ln\left | \frac{2\sqrt{x^{2}-x+1}+2x-1}{\sqrt{3}} \right |+C$. But the answer key says that the answer should be $\sinh^{-1}\left ( \frac{2x-1}{\sqrt{3}} \right )+C$. Any a... | Completing the square will yield
$$
x^2 - x + 1 = \left(x-\frac{1}{2}\right)^2 + \frac{3}{4}
$$
Normally, we will let $u=x-\frac{1}{2}$. However it can also be solved by letting $x-\frac{1}{2}=\frac{\sqrt3}{2}\sinh t$ and $dx=\frac{\sqrt3}{2}\cosh t\ dt$ which yields
$$
\begin{align}
\int \frac{dx}{\sqrt{x^{2}-x+1}}&=\... | {
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"timestamp": "2023-03-29T00:00:00",
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Distance point on ellipse to centre I'm trying to calculate the distance of a certain point of an ellipse to the centre of that ellipse:
The blue things are known: The lengths of the horizontal major radius and vertical minor radius and the angle of the red line and the x-axis. The red distance is the desired result. ... | I'm assuming that the centre is located on the origin.
Let us call the point where the red line meets the ellipse P. Let P have coordinates $(x_1, y_1) $. Notice that $tan\theta = y_1/x_1$. Let the horizontal semi-axis be $a$, meaning the ellipse is $2a$ wide. Similarly, let the vertical semi-axis be $b$.
Let the known... | {
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Prove that for infinitely many $n$, the numbers $(a+n)$,$(b+n)$,$(c+n)$ are mutually prime The problem is -
Let $a$, $b$ and $c$ be three distinct integers such that they are mutually prime. Show that for infinitely many $n$, the numbers $(a+n)$,$(b+n)$,$(c+n)$ are mutually prime.
I have been able to show that for in... | We prove that if $a$, $b$, and $c$ are distinct, then there are infinitely many $n$ such that $a+n$, $b+n$, and $c+n$ are pairwise relatively prime. We do not need to assume that $a$, $b$ and $c$ are pairwise relatively prime.
Without loss of generality we may assume that $a=0$ and $b$ and $c$ are positive, with $b\lt... | {
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Tricky Triangle Area Problem This was from a recent math competition that I was in. So, a triangle has sides $2$ , $5$, and $\sqrt{33}$. How can I derive the area? I can't use a calculator, and (the form of) Heron's formula (that I had memorized) is impossible with the$\sqrt{33}$ in it. How could I have done this? The ... | Use the $\frac{1}{4}\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}$ form of Herons' formula.
$$\begin{align}
& \frac{1}{4}\sqrt{4\cdot4\cdot25-(4+25-33)^2} \\
= & \frac{1}{4}\sqrt{4^2\cdot25-4^2} \\
= & \sqrt{25-1} \\
= & 2\sqrt{6}
\end{align}$$
| {
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"answer_id": 2
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$n$ is a square and a cube $a^2 = n = b^3\Rightarrow n\equiv 0,1\pmod{7}$
Verify that if an integer is simultaneously a square and a cube, then it must be either of the form ${7k}$ or ${7k +1}$.
I have no idea on how to proceed.
| Check the integers mod $7$. Which are squares and which are cubes:
$$0^2=0^3=0$$
$$1^2=1^3=1$$
$$2^2 = 4,\qquad 2^3=8\equiv 1$$
$$3^2 = 9 \equiv 2,\quad3^3=27\equiv 6$$
$$4^2 = 16 \equiv 2,\quad 4^3= 64 \equiv 1$$
$$5^2 = 25 \equiv 4,\quad 5^3= 125 \equiv 6$$
$$6^2 = 36 \equiv 1,\quad 6^3=216\equiv 6.$$
So only $0, 1, ... | {
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Limit as x approches infinity - Trouble with calculus I have this problem on a practice exam:
$\displaystyle \lim_{x\to\infty} 3x - \sqrt{9x^2+2x+1}$
We are dealing with L'hospitals rule, so when you plug $\infty$ in for $x$ you get
$\infty - \infty$. I multiplied by the conjugate to get:
$\displaystyle \frac{-2x-1}{3... | This is to answer "what is your thought process" (I cannot read LAcarguy's mind so I cannot say what his thought process is). The trick of multiplying and dividing by the reciprocal of a leading term is a standard trick when we have a limits going to infinity. For instance suppose that we have the following limit, $$\... | {
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How to find the values of a and b? If the polynomial 6x4 + 8x3 - 5x2 + ax + b is exactly divisible by the polynomial 2x2 - 5, then find the values of a and b.
| You can just write $$6x^4 + 8x^3 - 5x^2 + ax + b=(2x^2-5)(Ax^2+Bx+C)$$ Expand the rhs to get $$6x^4 + 8x^3 - 5x^2 + ax + b=2Ax^4+2Bx^3+(2C-5A)x^2-5Bx-5C$$ Now identify the coefficients of the differents powers of $x$. Using $x^4,x^3,x^2$, this leads to $A=3$, $B=4$,$C=5$.
I am sure that you can take from here.
| {
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is 121 divides the my pattern for base 2? Is $121|2^{120}-1$? If yes, is there any online free calculation to check these type of values?
Advanced thanks to one and all!
-Richard Sieman
| You can rewrite your expression as$$2^{120}-1\\=(2^{60}+1)(2^{60}-1)\\=(2^{60}+1)(2^{30}+1)(2^{30}-1)\\=(2^{60}+1)(2^{30}+1)(2^{15}+1)(2^{15}-1)$$
so $$2^{120}-1\pmod{ 121}\\\equiv(2^{60}+1)(2^{30}+1)(2^{15}+1)(2^{15}-1)\pmod{ 121}$$
but since $2^{15}-1=32767 \equiv 97 \pmod{ 121}$
$$2^{120}-1\pmod{ 121}\\\equiv(2^{60}... | {
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A closed form for the infinite series $\sum_{n=1}^\infty (-1)^{n+1}\arctan \left( \frac 1 n \right)$ It is known that $$\sum_{n=1}^{\infty} \arctan \left(\frac{1}{n^{2}} \right) = \frac{\pi}{4}-\tan^{-1}\left(\frac{\tanh(\frac{\pi}{\sqrt{2}})}{\tan(\frac{\pi}{\sqrt{2}})}\right). $$
Can we also find a closed form for th... | This is not an answer, but a useful way to transform the expression and link it to another, more simple sum:
$$\sum_{n=1}^{\infty} (-1)^{n+1} \arctan \frac{1}{n}=\frac{\pi}{4}-\sum_{n=1}^{\infty} \left( \arctan \frac{1}{2n}-\arctan \frac{1}{2n+1}\right)$$
$$\arctan \frac{1}{2n}-\arctan \frac{1}{2n+1}=\arctan \frac{1}{4... | {
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Given $g(1) = 6$, $g'(1) = -1$, find $d/dx(2 g(x)/(x^2 + 1))$ when $x = 1$ Why is the answer $-7$? I plugged $1$ into the equation and I ended up with $12/2$ and got $6$. Can someone explain to me what I did wrong?
| Applying Quotient Rule, we find that:
\begin{align*}
\frac{d}{dx}\left[\frac{2g(x)}{x^2+1} \right]
&= \frac{(x^2+1)\frac{d}{dx}[2g(x)] - 2g(x)\frac{d}{dx}[x^2+1]}{(x^2+1)^2} \\
&= \frac{(x^2+1)[2g'(x)] - 2g(x)[2x]}{(x^2+1)^2} \\
&= \frac{2(x^2+1)g'(x) - 4xg(x)}{(x^2+1)^2} \\
\end{align*}
Hence, substituting $x = 1$, we... | {
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Prove $\sin \frac{\alpha}{2}\sin \frac{\beta}{2}\sin \frac{\gamma}{2}\leq \frac {1}{8}$, $\alpha, \gamma\, \beta$ being angles of a triangle Prove $\sin \frac{\alpha}{2}\sin \frac{\beta}{2}\sin \frac{\gamma}{2}\leq \frac {1}{8}$
I defined $f(x,y,z)=\sin \frac{\alpha}{2}\sin \frac{\beta}{2}\sin \frac{\gamma}{2}$, and wa... | Using Algebra only,
$$2\sin\frac\alpha2\sin\frac\beta2=\cos\frac{\alpha-\beta}2-\cos\frac{\alpha+\beta}2$$
Now $\displaystyle\cos\frac{\alpha+\beta}2=\cdots=\sin\frac\gamma2$
Let $\displaystyle y=2\sin\frac\alpha2\sin\frac\beta2\sin\frac\gamma2$
$$\implies y=\left(\cos\frac{\alpha-\beta}2-
\sin\frac\gamma2\right)\sin\... | {
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Prove the identity $\frac{\cos B}{1-\tan B} + \frac{\sin B}{1-\cot B}=\sin B+\cos B$ I have worked on this identity from both sides of the equation and can't seem to get it to equal the other side no matter what I try.
$\displaystyle\frac{\cos B}{1-\tan B} + \frac{\sin B}{1-\cot B} =\sin B+\cos B$
| What I would do here is to first convert $\tan B$ and $\cot B$ into $\frac{\sin B}{\cos B}$ and $\frac{\cos B}{\sin B}$, respectively.
$$\frac{\cos B}{1-\tan B}+\frac{\sin B}{1-\cot B}$$
$$=\frac{\cos B}{\left(1-\dfrac{\sin B}{\cos B}\right)}+\frac{\sin B}{\left(1-\dfrac{\cos B}{\sin B}\right)}$$
Simplify the denominat... | {
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Equilateral triange, sum... Just a short question:
In a triangle we have $\sum \left(\frac{a}{b+c}\right)^{2}$. Is the triangle equilateral?
I have derived
| For all real numbers $x,y,z$ we have that
$3(x^2+y^2+z^2)\geq (x+y+z)^2$. (It follows from expanding $\sum (x-y)^2 \geq 0$)
Applying this to your sum, we have that
$\sum \left(\frac{a}{b+c}\right)^2 \geq \frac{1}{3}\left(\sum\frac{a}{b+c}\right)^2$ with equality iff $\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}$.
Now Nesb... | {
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How did Euler realize $x^4-4x^3+2x^2+4x+4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$? How did Euler find this factorization?
$$\small x^4 − 4x^3 + 2x^2 + 4x + 4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$$
where $\alpha = \sqrt{4+2\sqrt{7}}$
I know that he had s... | The other answers are great but were pure speculation since Euler published how he solves quartics.
From Elements of Algebra by Euler section 4 chapter 15, his new method for resolving fourth order equations. Additionally, Google books has a full copy to download or read online here.
Suppose the root of the equation i... | {
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To solve $ \frac {dy}{dx}=\frac 1{\sqrt{x^2+y^2}}$ How do we solve the differential equation $ \dfrac {dy}{dx}=\dfrac 1{\sqrt{x^2+y^2}}$ ?
| $\dfrac{dy}{dx}=\dfrac{1}{\sqrt{x^2+y^2}}$
$\dfrac{dx}{dy}=\sqrt{x^2+y^2}$
Apply the Euler substitution:
Let $u=x+\sqrt{x^2+y^2}$ ,
Then $x=\dfrac{u}{2}-\dfrac{y^2}{2u}$
$\dfrac{dx}{dy}=\left(\dfrac{1}{2}+\dfrac{y^2}{2u^2}\right)\dfrac{du}{dy}-\dfrac{y}{u}$
$\therefore\left(\dfrac{1}{2}+\dfrac{y^2}{2u^2}\right)\dfrac{d... | {
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Find the product $(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)(1-a_6)$ Let $1,$ $a_i$ for $1 \leq i \leq 6$ be the different roots of $x^7-1$.
Then find the product:
$(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)(1-a_6)$
I don't know how to proceed.
| We have $$(x-1)(x-a_1)(x-a_2)(x-a_3)(x-a_4)(x-a_5)(x-a_6)=(x-1)(1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}).$$ Hence we must have $$(x-a_1)(x-a_2)(x-a_3)(x-a_4)(x-a_5)(x-a_6)=1+x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}.$$ This is because I am dividing the same monomial $x-1$ from both sides. Hence we have $$\prod_{i=1}^{6}(1-a_i)=7.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/786469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Elevator Probability Question There are four people in an elevator, four floors in the building, and each person exits at random. Find the probability that:
a) all exit at different floors
b) all exit at the same floor
c) two get off at one floor and two get off at another
For a) I found $4!$ ways for the passengers... | Solutions using conditional probabilities:
Let's enumerate people $P_1, P_2, P_3, P_4$.
a) all exit at different floors
$P_1$ can pick any of 4 floors with probability $\frac{1}{4}$ each, so we have 4 cases. Because all 4 cases are equivalent we focus on just one. Now, given the floor $P_1$ exited at, $P_2$ can exit on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/786708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 2
} |
find the point $P$ such that the expression has minimum value Let $ABC$ be a triangle with sides
$$a,b,c.$$
Find a point $P$ inside the triangle such that
$$a(PA)^2+b(PB)^2+c(PC)^2$$
is minimum
| Let $I$ be the incenter of $ABC$. $I$ is the barycenter of the weighted points $(A;a)$,$(B;b)$ and $(C;c)$. This means that
$$
a\overrightarrow{IA}+b\overrightarrow{IB}+c\overrightarrow{IC}=\overrightarrow{0}\tag{1}
$$
Now
$$\eqalign{
a(PA)^2+b(PB)^2+c(PC)^2&=a(\overrightarrow{IA}-\overrightarrow{IP})^2
+a(\overrightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/787815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find minimum value of the expression x^2 +y^2 subject to conditions Find the values of $x,y$ for which $x^2 + y^2$ takes the minimum value where $(x+5)^2 +(y-12)^2 =14$.
Tried Cauchy-Schwarz and AM - GM , unable to do.
| Any point satisfying $\displaystyle(x+5)^2 +(y-12)^2 =14$ can be expressed as $\sqrt{14}\cos\phi-5,\sqrt{14}\sin\phi+12$
$\displaystyle x^2 + y^2=14(\cos^2\phi+\sin^2\phi)+2\sqrt{14}(12\sin\phi-5\cos\phi)+5^2+12^2$
$\displaystyle=14+12^2+5^2+2\sqrt{14}(12\sin\phi-5\cos\phi)$
This will attain minimum if $\displaystyle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/787894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Evaluating Combination Sum $\sum{n+k\choose 2k} 2^{n-k}$ Evaluate $$\sum_{k=0}^n{n+k\choose 2k} 2^{n-k}$$
So im not really sure how to begin with this. I would imagine we start with dividing out $2^{n}$, but not really sure much past that
| The method used here is that of the generating function. Let $S_{n}$ be the series to be summed
\begin{align}
S_{n} = \sum_{k=0}^{n} \binom{n+k}{2k} \ 2^{n-k}.
\end{align}
The generating function and its reduction are as follows.
\begin{align}
\sum_{n=0}^{\infty} S_{n} \frac{t^{n}}{2^{n}} &= \sum_{n=0}^{\infty} \sum_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/792567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Finding determinant for matrix using upper triangle method Here is an example of a matrix, and I'm trying to evaluate its determinant:
$$
\begin{pmatrix}
1 & 3 & 2 & 1 \\
0 & 1 & 4 & -4 \\
2 & 5 & -2 & 9 \\
3 & 7 & 0 & 1 \\
\end{pmatrix}
$$
When applying first row operation i get:
$$
\begin{pmatrix}
1 & 3 & 2 & 1 \\
0 ... | If we multiply a certain row or column of a matrix $A$ by some scalar $\lambda$ then determinant of $A$ changes to $\lambda|A|.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/793217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What remainder does $34!$ leave when divided by $71$? What is the remainder of $34!$ when divided by $71$?
Is there an objective way of solving this?
I came across a solution which straight away starts by stating that
$69!$ mod $71$ equals $1$ and I lost it right there.
| Continuing in the line of Samrat Mukhopadhyay's answer, a method that is hardly any easier than actually computing $34!\pmod{71}$ by simply multiplying factor by factor:
By Wilson's theorem we know that $70!\equiv-1\pmod{71}$, from which it follows that
$$(34!)^2\times35\times36\equiv34!\times36!\equiv70!\equiv-1\pmod{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/794470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
} |
Finding eigenvectors of a matrix I want to find all eigenvalues and eigenvectors of the matrix $\begin{bmatrix}0&1&0\\0&0&1\\-1&0&0\end{bmatrix}$.
Here is how I find eigenvalues:
$$\begin{align*}
\det(A - \lambda I) &= \det \Bigg(\begin{bmatrix}0&1&0\\0&0&1\\-1&0&0\end{bmatrix} - \begin{bmatrix}\lambda&0&0\\0&\lambda... | Since your characteristic equation is:
$$
\lambda^3 = -1 \rightarrow \lambda = e^{\pi i + \frac{2n\pi}{3}i}
$$
and gives three distinct eigenvalues, there are exactly three eigenvectors only one of which has eigenvalue $\lambda = -1$.
$$
\begin{bmatrix} 1 & 1 & 0 \\
0 &1& 1 \\
-1 &0 & 1
\end{bmatrix} \rightarrow \begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/795256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Finding surface area of part of a plane that lies inside a cylinder??? I have a question::
Let $S$ be the part of plane $x+2y+3z=1$ that lies inside cylinder $x^2 + y^2 = 3$
They want me to find the surface area of S??
This is a way harder question than all my previous ones, and I think I should start by finding the in... | With the surface defined by $g(x,y,z)=x+2y+3z-1=0$ over the domain $(x,y)\in C=\{(a,b):a^2+b^2\le3\}$, use the formula:
\begin{align}
\text{Surface Area} &= \int \int_C \sqrt{\frac{g_x^2+g_y^2+g_z^2}{g_z^2}}\mathrm{d}x \mathrm{d}y\\
&=\int \int_C \sqrt{\frac{1^2+2^2+3^2}{3^2}}\mathrm{d}x \mathrm{d}y\\
&=\int \int_C \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/798938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Algebraic solution for the intersection point(s) of two parabolas I recently ran through an algebraic solution for the intersection point(s) of two parabolas $ax^2 + bx + c$ and $dx^2 + ex + f$ so that I could write a program that solved for them. The math goes like this:
$$
ax^2 - dx^2 + bx - ex + c - f = 0 \\
x^2(a -... | You should recognise a form of the quadratic formula:$$(a-d)x^2+(b-e)x+(c-f)=0$$
which gives $$x=\frac {-(b-e)\pm \sqrt {(b-e)^2-4(a-d)(c-f)}}{2(a-d)}$$
This is the same as yours except for a missing factor of $\frac 14$ under your square root, which you lost when you took the square root near the end.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/799519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange Multipliers
Find the extrema of $x+y+z$ subject to $x^2 - y^2 = 1$ and $2x + z = 1$ using Lagrange multipliers.
So I set it up:
$$
1 = 2x\lambda_1 + 2\lambda_2 \\
1 = -2y\lambda_1 \\
1 = \lambda_2
$$
Plug in for $\lambda_2$:
$$
1 = 2x\lambd... | Note: There is no extremas with the current situation...So I slightly changed one of the
"constraints" to:
$x^2 - 2y^2 = 1$, and to show you how to solve if using one parameter $LM$ method.
$x + y + z = x + y + (1 - 2x) = y - x + 1$, subject to: $x^2 - 2y^2 = 1$.
Define $f(x,y) = -x + y + 1$, and $g(x,y) = x^2 - 2y^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/800575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find matrix determinant How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon form given this is all I know and there exists no additional information.
$\left[
\begin{arra... | If you know how value of determinant is influenced by elementary row/column operations (see ProofWiki) then you could start by adding all other columns to the last one (which does not change the determinant) and the rest is relatively easy:
$$
\begin{vmatrix}
1+x & 2 & 3 & 4 \\
1 & 2+x & 3 & 4 \\
1 & 2 & 3+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/802517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
integral involving hypergeometric function $\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{y})}{y}\,dy$ I arrived at the following result
$$\tag{1}\int^\infty_0 z^{p-1} E^2(z)\,dz=\frac{\Gamma(p)}{p}\int^1_0\frac{_2F_1(p,p;p+1;-\frac{1}{z})}{z}\,dz$$
where the exponential integral $E(z)$ is defined as
$$E(z)=\int^\infty_z \fr... | Let
\begin{align}
E_{1}(x) = \int_{x}^{\infty} \frac{e^{-t}}{t} \, dt
\end{align}
then the following two identities can be seen to be
\begin{align}
\int_{0}^{\infty} x^{n} E_{1}(ax) E_{1}(bx) \, dx &= - \frac{n!}{n+1} \left[ \frac{1}{a^{n+1}} \left\{ \ln\left(\frac{b}{a+b}\right) + \sum_{m=1}^{n} \frac{1}{m} \left( \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Proof for $\sin(x) > x - \frac{x^3}{3!}$ They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried:
$\sin(x) + x -\frac{x^3}{6} > 0 \\$
then I computed the derivative of that function to d... | Observe that:
$\\ \\ \displaystyle \sin(3\gamma)=\sin(2\gamma)\cos(\gamma)+\sin(\gamma)\cos(2\gamma)=2\sin(\gamma)\cos^2(\gamma)+\sin(\gamma)(1-2\sin^2(\gamma))=2\sin(\gamma)(1-\sin^2\gamma)+\sin(\gamma)(1-2\sin^2(\gamma))=3\sin(\gamma)-4\sin^3(\gamma)\Rightarrow \sin^3(\gamma)=\frac{1}{4}\left(3\sin(\gamma)-\sin(3\gam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 9,
"answer_id": 2
} |
How to find determinant of this matrix? Is there a manual method to find $\det\left(XY^{-1}\right)$ ?
Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\
1 & 3 & 3^2 & \cdots & 3^{2012} \\
1 & 4 & 4^2 & \cdots & 4^{2012} \\
\vdots & \vdots & \vdots & \cdots & \vdots \\
1 & 2014 & 2014^2 & \cdo... | Not a solution, but a possible step in the right direction (too long for a comment):
Following Cramer's rule, we note that the solution $\vec y = (y_1,\dots,y_{2014})^T$ to
$$X \,\vec y = \pmatrix{
\frac{2^2}{4} & \frac{3^2}{5} & \dfrac{4^2}{6} & \cdots & \dfrac{2014^2}{2016}
}^T$$
Will satisfy
$$
y_1 = \det(Y^T)/\det... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Prove that $x^3 + x^3 y^2 + x^2 y^3 + x^2 + y^2 + y^3 \geq 2xy(x+y+xy)$ Prove that $x^3 + x^3 y^2 + x^2 y^3 + x^2 + y^2 + y^3 \geq
2xy(x+y+xy)$ for $x,y \in \mathbb{R}^+$.
I started by multiplying everything out on the RHS to get the equivalent statement
\begin{align*}
x^3 + x^3 y^2 + x^2 y^3 + x^2 + y^2 + y^3 \ge... | By $AM-GM :$
$$x^3+x^3y^2\ge 2x^3y$$
$$x^2+y^2\ge2xy$$
$$y^3 +x^2y^3\ge2xy^3$$
Add :
$$\text{LHS}\ge2xy(x^2+y^2+1)$$
WLOG Assume $x\ge y\ge 1$
Use rearrangement inequality on same inequality :
$$x^2+y^2+1\ge x+y+xy$$
By transitivity...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/807654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Examples of quasigroups with no identity elements If you scroll to the bottom of this page, there is a table claiming quasigroups have divisibility but not identity (in general).
What would be some examples of quasigroups without an identity element?
| The Cayley tables:
$$\begin{array}{c|ccccccc}
\ast & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline
1 & 1 & 3 & 2 & 5 & 4 & 7 & 6\\
2 & 3 & 2 & 1 & 6 & 7 & 4 & 5\\
3 & 2 & 1 & 3 & 7 & 6 & 5 & 4\\
4 & 5 & 6 & 7 & 4 & 1 & 2 & 3\\
5 & 4 & 7 & 6 & 1 & 5 & 3 & 2\\
6 & 7 & 4 & 5 & 2 & 3 & 6 & 1\\
7 & 6 & 5 & 4 & 3 & 2 & 1 & 7
\end{array... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/808122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Algebra Manipulation Contest Math Problem The question was as follows:
The equations $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ have two roots in common. Compute the product of these common roots.
Because $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ it means that $x^3+Ax+10=x^3+Bx^2+50$
Take $x^3+Ax+10=x^3+Bx^2+50$ and remove $x^3$ from ... | Let $a,b$ be the roots.
Then $a,b$ are roots of
$$x^3+Ax+10=0$$
The sum of the three roots of this polynomial is negative the coefficient of $x^2$, thus $0$. It follows that the third root is $-(a+b)$.
As the product of the three roots is $-10$ we get
$$ab(a+b)=10$$
Now let $c$ be the third root of
$$x^3+Bx^2+50=0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Elliptic integral $\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk$ Question:
Prove that
$$\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk=\frac{1}{16\pi}\Gamma^4\left(
\frac{1}{4}\right)$$
My attempt
Start by the transformation
$$k \to \frac{2\sqrt{k}}{1+k}$$
Hence we have
$$\int^{1}_0 K\left(\frac{2\sqrt{k}}{1+k}\right)\,\frac{1... | A simple method for the last integral:
\begin{align}
\int \limits_0^1 \frac{\operatorname{K}(k)}{\sqrt{k}} \, \mathrm{d} k &= \int \limits_0^1 \int \limits_0^{\pi/2} \frac{\mathrm{d} \phi}{\sqrt{k(1-k^2 \sin^2(\phi))}} \, \mathrm{d} k \stackrel{\text{Tonelli}}{=} \int \limits_0^{\pi/2} \int \limits_0^1 \frac{\mathrm{d}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
} |
Integer solutions to $x^x=122+231y$ How can I find the integer solutions to the following equation (without a script or trial and error)?
$$x^x=122+231y$$
| By the Chinese Remainder Theorem, to know the value of $x^x$ modulo $231$ we only need to know it modulo $3,7,11$.
The value of $x^y$ modulo $n$ depends on the values of $x$ modulo $n$ and $y$ modulo $\phi(n)$.
So if $n$ is prime, the value of $x^x$ modulo $n$ depends on $x$ modulo $n(n-1)$.
A computation gives that
$x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/813736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
How do I find the sum of the infinite geometric series? $$2/3-2/9+2/27-2/81+\cdots$$
The formula is $$\mathrm{sum}= \frac{A_g}{1-r}\,.$$
To find the ratio, I did the following:
$$r=\frac29\Big/\frac23$$
Then got:
$$\frac29 \cdot \frac32= \frac13=r$$
and $$A_g= \frac23$$
Then I plug it all in and get:
$$\begin{align*}
\... | $$\frac{2}{3}-\frac{2}{9}+\frac{2}{27}-\frac{2}{81}+...$$
$$\frac{2}{3}+\frac{2}{27}+\cdots - (\frac{2}{9}+\frac{2}{81}+\cdots)$$
$$2(\dfrac{\frac{1}{3}}{1-\frac{1}{9}}) - 2(\dfrac{\frac{1}{9}}{1-\frac{1}{9}})$$
$$2.(\frac{3}{8}) - 2(\frac{1}{8})$$
$$\frac{1}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/816306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
} |
How to solve $\int \frac{\,dx}{(x^3 + x + 1)^3}$?
How to solve $$\int \frac{\,dx}{(x^3 + x + 1)^3}$$ ?
Wolfram Alpha gives me something I am not familiar with. I thought that the idea was using partial fractions because $x^3$ and $x$ are bijections, there must be a real root but it seems that Wolfram Alpha is using... | The integral is a rational function so one ought to be able to apply the method of partial fractions, however, the real root of $x^3+x+1$ is irrational without a nice expression, this makes it difficult to apply partial fractions, however some progress can be made.
Consider first what kinds of functions we get from int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Get variables with Matrix I try to get the variables for this equation:
$$\begin{cases}
6x_1 + 4x_2 + 8x_3 + 17x_4 &= -20\\
3x_1 + 2x_2 + 5x_3 + 8x_4 &= -8\\
3x_1 + 2x_2 + 7x_3 + 7x_4 &= -4\\
0x_1 + 0x_2 + 2x_3 -1x_4 &= 4
\end{cases}$$
So i started with:
$$ \begin{pmatrix}
6 & 4 & 8 & 17 & -20 \\
... | $$ \begin{pmatrix}
6 & 4 & 8 & 17 & -20 \\
0 & 0 & -2 & \color{red}{-1} & -4 \\
0 & 0 & -6 & 3 & -12\\
0 & 0 & 2 & -1 &4\\
\end{pmatrix}$$
should be $$ \begin{pmatrix}
6 & 4 & 8 & 17 & -20 \\
0 & 0 & -2 & \color{green}{+1} & -4 \\
0 & 0 & -6 & 3 & -12\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
limits of function without using L'Hopital's Rule $\mathop {\lim }\limits_{x \to 1} \frac{{x - 1 - \ln x}}{{x\ln x+ 1 - x}} = 1$ Good morning.
I want to show that without L'Hopital's rule :
$\mathop {\lim }\limits_{x \to 1} \frac{{x - 1 - \ln x}}{{x\ln x + 1 - x}} = 1$
I did the steps
$
\begin{array}{l}
\mathop {\lim... | $$
\displaylines{\mathop {\lim }\limits_{_{x \to 0} } \frac{{e^x - x - 1}}{{x^2 }} = \frac{1}{2} \cdots \left( 1 \right) \cr}
$$
$$
\displaylines{ \mathop {\lim }\limits_{_{t \to 0} } \left( {\frac{{t^2 }}{{te^t - e^t + 1}}} \right) = 2 \cdots \left( 2 \right) \cr}
$$
$$\displaylines{ \mathop {\lim }\limits_{_{x \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
In the Quadratic Formula, what does it mean if $b^2-4ac>0$, $b^2-4ac<0$, and $b^2-4ac=0$? Concerning the Quadratic Formula:
What does it mean if $b^2-4ac>0$, $b^2-4ac<0$, and $b^2-4ac=0$?
| Since one has the term $\sqrt{b^2-4ac}$ in solution of the quadratic formula, if $b^2-4ac>0$, then the equation has two real solutions (since $\sqrt{b^2-4ac}$ is real, and $b$ and $2a$ are also real). When $b^2-4ac<0$, then the quadratic equation has two complex solutions (since $\sqrt{b^2-4ac}$ is complex imaginary). ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Prove $a_n=1+\frac{a_{n-1}}{1+a_{n-1}}$ increasing There is a homework question in Calculus-1 course:
Calculate the limit of $\{a_n\}$: $$a_1=1,\ a_n=1+\frac{a_{n-1}}{1+a_{n-1}}$$
I think the key points are bounded and increasing, and I have proved that $$a_n\in(1, 2)$$
If I knew it's increasing then $$a=1+\frac{a}{1+... | Consider the sequence
\begin{align}
a_{n} = 1 + \frac{a_{n-1}}{1+a_{n-1}}
\end{align}
where $a_{1} = 1$.
Let $2 \alpha = 1 + \sqrt{5}$ and $2 \beta = 1-\sqrt{5}$. It is seen that $\alpha > \beta$ and $\beta^{n} \rightarrow 0$ as $n \rightarrow \infty$. Now, the terms of $a_{n}$ are $a_{n} \in \{ 1, 3/2, 8/5, \cdots \}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Volume between cylinder and plane Problem: Find the volume bounded by $z = y^2, x =0, y =0, z =9-x$.
My working:
$z$ goes from $y^2$ to $9-x$ so these are the limits of integration.
Work out the points of intersection of $9-x$ and $y^2$. When $y=0$, $9-x=0$ and $x=9$. So $x$ goes from 0 to 9. When $x=0$, $y^2 = 9$ so $... | Alternately the set up is:
$\displaystyle \int_{0}^3 \int_{0}^{9-y^2} \int_{y^2}^{9-x} 1dzdxdy = \dfrac{324}{5}$ (already checked)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The differential equation: $ \arctan (y) = \arctan(x)+C .$ I solved the equation and stalled. Help with decision please.
$$(1+y^2)\,dx=(1+y^2)\,dy \iff \int \frac{dx}{1+x^2} = \int\frac{dy}{1+y^2} $$
Transformed expression for the table of integrals.
$$ \arctan (y) = \arctan(x)+C $$
Prompt how to further transform ex... | The differential set
$$
(1+y^{2}) \,dx = (1+x^{2})\, dy
$$
can be integrated as seen by
\begin{align}
\int \frac{dy}{1+y^{2}} = \int \frac{dx}{1+x^{2}}
\end{align}
and leads to
\begin{align}
\tan^{-1}(y) = \tan^{-1}(x) + c
\end{align}
or
\begin{align}
y = \tan(\tan^{-1}(x) + c).
\end{align}
Now using
\begin{align}
\... | {
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"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Compute an integral with residue theorem Using residue theorem, compute the following integral:
$$
\int_{0}^{2\pi}\frac{\left( 1+2\cos t\right) ^{n}\cos\left( nt\right)
}{5+4\cos t}\operatorname*{dt}.
$$
Or a source with a solution.
| \begin{align} \int_{0}^{2 \pi} \frac{(1+2 \cos t)^{n} \cos (nt)}{5 + 4 \cos t} \ dt &= \text{Re} \int_{0}^{2 \pi} \frac{(1+2 \cos t)^{n} e^{int}}{5 + 4 \cos x} \ dt \\ &= \text{Re} \int_{0}^{2 \pi}\frac{(1+ e^{it} + e^{-it})^{n}e^{int}}{5 + 2(e^{it} + e^{-it})} \ dt \\ &= \text{Re} \int_{|z|=1} \frac{(1+z+z^{-1})^{n}z^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/829090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find all the asymptote of $1-x+\sqrt{2+2x+x^2}$ I find
$$\lim_{x\rightarrow\infty}1-x+\sqrt{2+2x+x^2}=2,$$ but i'am stuck when $x\rightarrow-\infty$ how to find that $y=-2x$ is an oblique asymptote. Any idea?
| General Method
Consider a curve given by the equation $f(x, y) = 0$, where $f(x, y)$ is a polynomial of degree $n$ (in $x$ and $y$). If it contains the term $y^n$ (with some non-zero coefficient), then it has no asymptote parallel to the $y$-axis. Otherwise, equate the total coefficient of the highest degree term of $y... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y
Show that every prime number in form $ p=4m+1 $ could be showed as $ p = x^2+y^2$ (x and y are natural)
... | *
*It can be: $$x=2n+1 \text{ and } y=2k+1 \ \text{ OR } \ x=2n \text{ and } y=2k \ \text{ OR } x=2n \text{ and } y=2k+1 \text{ OR } x=2n+1 \text{ and } y=2k$$
You can check each case and you will see that it cannot be $x^2+y^2=4m+3$
For example,at the first case you will get:
$$x^2+y^2=(2n+1)^2+(2k+1)^2=4n^2+4n+1+4k^... | {
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"url": "https://math.stackexchange.com/questions/830577",
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"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Understanding 2012 AMC 12B #23
Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that
$P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros
at $x=-59$,$-57$,$-51$ and $-49$. What is the sum of the minimum values of
$P(x)$ and $Q(x)$?
I found a solution here, and I was abl... | Claim: If $ \alpha, \beta$ are the roots of $ Q(x) = a + \sqrt{b}$ and $\gamma , \delta$ are the roots of $Q(x) = a - \sqrt{b}$, then
$$|\alpha - \beta | > | \gamma - \delta|.$$
Proof: This follows by considering the graph of the monic quadratic $y=Q(x)$ and the lines $ y = a + \sqrt{b}$ and $ y = a - \sqrt{b}$. Use ... | {
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"source": "stackexchange",
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A sum regarding prime factorization Prime factorization of $n$ is $n = p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$
Let $f(n) = \left((p_1^{a_1}+1)(p_2^{a_2}+1)(p_3^{a_3}+1)\cdots(p_k^{a_k}+1)\right)$
I want to find the value of $$\sum_{n=1}^{N}f(n)$$
For example if $N=6$, then the answer is $(1^{1}+1)+(2^{1}+1)+(3^{1}... | Let $F(N) = \sum_{n=1}^N f(n)$ where $f(n) = \sum_{d|n, \gcd(d,n/d) = 1} d$ as was noted in the comments by Hagen von Eitzen.
Now changing the order of summation and noting that $n$ is of the form $n = n'd$, we have
$$F(N) = \sum_{d=1}^N \sum_{\gcd(d,n') = 1, 1 \le n' d \le N} d,$$
which we can also write (by replacing... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/832779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$AM+GM+HM+RMS=66$ For two reals $a,b\in\mathbb{R}$ Arithmetic mean(AM), Geometric Mean(GM), Harmonic Mean(HM) and Root mean square (RMS) all are integers and
$AM+GM+HM+RMS=66$
Find all such $a,b$
I have assumed $a+b=u$ and $ab=v$ and written all the means in the terms of $u,v$ and used trial and error after some anal... | $a+b=2u,ab=v^2 \implies AM=u,GM=v.HM=\dfrac{2v^2}{2u}=\dfrac{v^2}{u},RMS=\sqrt{2u^2-v^2}$
$HM$ is integer ,$\implies u=tq^2, t $ is not a perfect square number $ v=tqp \implies RMS=\sqrt{2t^2q^4-t^2q^2p^2}=tq\sqrt{2q^2-p^2} \implies tq^2+tqp+tp^2+tq\sqrt{2q^2-p^2}=66 \implies t(q^2+pq+p^2+q\sqrt{2q^2-p^2})=66 $
$u \ge... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find the value of a + b + c + d Let $a$ and $b$ be the roots of the equation: $x^2 - 10cx - 11d = 0$ where $c$ and $d$ be the roots of $x^2 - 10ax - 11b = 0$. Find the value of $a+b+c+d$, assuming that they all are distinct.
I first tried making an equation with roots $(a+b)$ and $(c+d)$ to get the sum of the roots, ho... | My Solution ::
Given $a,b$ are the roots of $x^2-10cx-11d=0.$
So $$ a+b = 10c \tag{1}$$
and
$$ab = -11d \tag{2} $$
and $c,d$ are the roots of $x^2-10ax-11b=0.$
So $$c+d=10a\tag{3}$$
and
$$cd=-11b \tag{4}$$
So $$a+b+c+d = 10(a+c). \tag{5}$$
Now $$\frac{a+b}{c+d} = \frac{10c}{10a}=\frac{c}{a}\Rightarrow a^2+ab=c^2+cd\Rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/834745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 0
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Fair die being rolled repeatedly A fair die is rolled repeatedly, and let $X$ record the number of the roll when the 1st $6$ appears. A game is played as follows. A player pays \$1 to play the game. If $X\leq 5$ , then he loses the dollar. If $6 \le X \le 10$, then he gets his dollar back plus \$1. And if $X... | The probability the number is $X$ is $(\dfrac{5}{6})^{X-1}\cdot\dfrac{1}{6}$
Therefore the expected gains of a game is
$$\sum_{X=6}^{\infty}(\dfrac{5}{6})^{X-1}\cdot\dfrac{1}{6}+2\sum_{X=11}^\infty(\dfrac{5}{6})^{X-1}\cdot\dfrac{1}{6}=\dfrac{1}{6}(\sum_{X=6}^\infty(\dfrac{5}{6})^{X-1}+2\sum_{X=11}^\infty(\dfrac{5}{6})... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/836488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Check entry extended ternary Golay code The extended ternary Golay code is the linear $[12,6,6]$-code with the following generator matrix:
$$
C=\left(
\begin{array}
&1&0&0&0&0&0&0&1&1&1&1&1\\
0&1&0&0&0&0&1&0&1&2&2&1\\
0&0&1&0&0&0&1&1&0&1&2&2\\
0&0&0&1&0&0&1&2&1&0&1&2\\
0&0&0&0&1&0&1&2&2&1&0&1\\
0&0&0&0&0&1&1&1&2&2&1&0
... | Posting my comment as an answer lest this question gets stuck in the unanswered queue.
It is a possible explanation that the intended generator matrix for the code $C$ was
$$
G=\left(
\begin{array}
&1&0&0&0&0&0&0&1&1&1&1&1\\
0&1&0&0&0&0&2&0&1&2&2&1\\
0&0&1&0&0&0&2&1&0&1&2&2\\
0&0&0&1&0&0&2&2&1&0&1&2\\
0&0&0&0&1&0&2&2&2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/837122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove this identity... $$\frac{\sin 2x}{1+\cos 2x} \times \frac{\cos x}{1+\cos x}=\tan\frac{x}{2}$$
This is what I've done:
$$\frac{2\sin x \cos x}{1+\cos^2 x-\sin^2 x} \times \frac{\cos x}{1+\cos x}=$$
$$\frac{2\sin x \cos x}{2\cos^2 x} \times \frac{\cos x}{1+\cos x}=$$
$$\frac{\sin x}{1+\cos x}$$
I have no idea what ... | By your estimate we have $$\frac{\sin x}{1+\cos x}=\frac{2\sin \frac{x}{2}\cos \frac{x}{2}}{2\cos^2\frac{x}{2}}=\tan\frac{x}{2}.$$
| {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Solving $x-\sqrt{(x^2-36)} = {(x-6)^2\over 2x+12}$. I have problem with this equation:
$$x-\sqrt{(x^2-36)} = {(x-6)^2\over 2x+12}$$
Any ideas on beatiful solving?
| Assume $x+6=a$ and $x-6=b$
So LHS= $(a+b)/2-\sqrt{ab}=1/2 * (\sqrt a-\sqrt b)^2$.
RHS= $b^2/2a$.
Hence, $b/\sqrt a=\sqrt a- \sqrt b$
Hence $b=a-\sqrt{ab}$
Therefore $\sqrt {ab}=a-b=12.$ Now, put $a=12+b$ and solve $b(12+b)=144$.
Hope this helps. Correct me if I'm wrong!
| {
"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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Probability Help (die problem) A die is rolled 20 times. How many different sequences
a) each number 1-6 is rolled exactly three times
My Answer: (20 choose 6)*(3 choose 1)
b) each number 1-6 are each rolled exactly once in the first six rolls?
My Answer: (20 choose 6)*(6 choose 1)
c) each number rolled is at least as... | a) I am considering it to mean atleast 3 times, which can be obtained by an exponential generating function:
\begin{align*}
G(x) &= \left(\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}\right)^6 \\
\left[\frac{x^{20}}{20!}\right]G(x) &= \frac{11}{414720}\cdot 20! = 64530097632000
\end{align*}
b) has been answered : $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Product in terms of $n$ of $\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$ What is the following product in terms of $n$?
$$\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$$
Thank you.
| $$\frac{1}{2}\frac{3}{4}\frac{5}{6} \dotsb \frac{2n-1}{2n}=\frac{1\mathbf{2}3\mathbf{4}5 \dotsb (2n-1)\mathbf{2n}}{(2 \cdot 4 \cdot 6 \dotsb 2n)^2}=\frac{(2n)!}{2^{2n}(n!)^2}$$
| {
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"url": "https://math.stackexchange.com/questions/842911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Proving $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$
Prove $\frac2\pi x \le \sin x \le x$ for $x\in [0,\frac {\pi} 2]$.
I tried to do this in two ways, I'm not sure about CMVT and I have a problem with the other way.
Using Cauchy's MVT:
RHS:
$\sin x \le x \implies \frac {\sin x}{x}\le 1$
So define: ... | For any $x \in (0,\frac{\pi}{2})$, consider the expression
$$\frac{\sin x - \sin 0}{x - 0} - \frac{\sin\frac{\pi}{2} - \sin x}{\frac{\pi}{2}- x}
= \frac{\sin x}{x} - \frac{1-\sin x}{\frac{\pi}{2} - x}\tag{*1}
$$
Apply MVT on for the first term on $[0,x]$ and the second term on $[x,\frac{\pi}{2}]$, we can find two numbe... | {
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"url": "https://math.stackexchange.com/questions/842978",
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"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
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Basic Trigonometry Question If $\cos{(A-B)}=\frac{3}{5}$ and $\sin{(A+B)}=\frac{12}{13}$, then find $\cos{(2B)}$.
Correct answer = 63/65.
I tried all identities I know but I have no idea how to proceed.
| From
$$\cos{(A-B)}=\frac{3}{5}\Rightarrow \sin(A-B)=\sqrt{1-(3/5)^2}=\frac{4}{5}$$
and from
$$\sin{(A+B)}=\frac{12}{13}\Rightarrow \cos(A+B)=\sqrt{1-(12/13)^2}=\frac{5}{13}$$
then
$$\cos(2B)=\cos((A+B)-(A-B))=$$
$$=\cos(A+B)\cos(A-B)+\sin(A+B)\sin(A-B)=$$
$$=\frac{5}{13}\frac{3}{5}+\frac{12}{13}\frac{4}{5}=\frac{63}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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how to solve this question of polynomials
Given the polynmial is exactly divided by $x+1$, when it is divided by
$3x-1$, the remainder is $4$. The polynomial leaves remainder $hx+k$ when divided by
$3x^2+2x-1$. Find $h$ and $k$.
This is the question which is confusing me.. i have done this question like this:
$p(... | The error in your solution is that p(−1)=0 implies h(−1)+k=0, not 4,Right
| {
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"source": "stackexchange",
"question_score": "1",
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Meaningful lower-bound of $\sqrt{a^2+b}-a$ when $a \gg b > 0$. I know that, for $|x|\leq 1$, $e^x$ can be bounded as follows:
\begin{equation*}
1+x \leq e^x \leq 1+x+x^2
\end{equation*}
Likewise, I want some meaningful lower-bound of $\sqrt{a^2+b}-a$ when $a \gg b > 0$.
The first thing that comes to my mind is $\sqrt{a... | By the mean value theorem,
$$\sqrt{1 + x} - 1 = f(1 + x) - f(1) = x f'(c)$$
where $f$ is square root, $f'$ is its derivative,
and $c$ is some point in $[1, 1+x]$.
We need a lower bound and $f'$ is decreasing,
so $c$ is at worst $1 + x$ and we obtain
$$x f'(c) ≥ xf'(1 + x) = \frac{x}{2\sqrt{1 + x}}.$$
Backtrack: you wan... | {
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"source": "stackexchange",
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What technique turns $\frac{x}{(x-2)^2(x+1)}$ into $-\frac{1}{9x+9}+\frac{1}{9x-18}+\frac{2}{3(x-2)^2}$? I found this:
Let's rewrite the integrand so that it's easier to integrate:
$$\dfrac{x}{(x-2)^2(x+1)} = -\dfrac{1}{9x+9}+\dfrac{1}{9x-18}+\dfrac{2}{3(x-2)^2}$$
I see how the expressions are equal, but I don't kn... | The technique that is being used is called partial fraction decomposition. You'll get "tons" of hits if you Google the phrase.
As applied to this particular question, first note that
$$\dfrac{x}{(x-2)^2(x+1)} = -\dfrac{1}{9x+9}+\dfrac{1}{9x-18}+\dfrac{2}{3(x-2)^2}$$
$$= -\frac 19\cdot \frac 1{x + 1} + \frac 19 \cdot \f... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}$ is a strictly decreasing function. This is part of an actuarial science problem. Unfortunately, the official solution of this problem takes the derivative of
$$\dfrac{0.5x^2 + x + 1}{x^2 + x + 1}\text{, } \quad x \geq 0\text{.}$$
and shows that it is always $\leq 0$. Ho... | We can write the function as $\dfrac{0.5x^2+x+1}{x^2+x+1} = 1 - \dfrac{0.5x^2}{x^2+x+1} = 1 - \dfrac{0.5}{1+\dfrac{1}{x}+\dfrac{1}{x^2}}$.
You can easily see that $1+\dfrac{1}{x}+\dfrac{1}{x^2}$ is strictly decreasing, so $\dfrac{0.5}{1+\dfrac{1}{x}+\dfrac{1}{x^2}}$ is strictly increasing.
Therefore, $1 - \dfrac{0.5... | {
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"source": "stackexchange",
"question_score": "7",
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How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$ How to prove $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty.$$
I try to do like $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=\sum_{N=1}^\infty \sum_{n+m=N}^\infty \frac{1}{m^2+n^2}=\sum_{N=1}^\infty \sum_{m=1}^{N-1... | If the sum were finite, then we could get a contradiction as follows. Breaking it up into 4 sums depending on whether or not $m$ and $n$ are even, we have
$$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2} = \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{(2m)^2+(2n)^2} + \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 1
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Evaluation of $ \lim_{x\rightarrow \infty}\left\{2x-\left(\sqrt[3]{x^3+x^2+1}+\sqrt[3]{x^3-x^2+1}\right)\right\}$
Evaluate the limit
$$
\lim_{x\rightarrow \infty}\left(2x-\left(\sqrt[3]{x^3+x^2+1}+\sqrt[3]{x^3-x^2+1}\right)\right)
$$
My Attempt:
To simplify notation, let $A = \left(\sqrt[3]{x^3+x^2+1}\right)$ and $B ... | As $x \to\infty$, $A\to x$ and $B\to x$. Therefore the expression would tend to $0$.
| {
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"url": "https://math.stackexchange.com/questions/851849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 4
} |
For the polynomial For the polynomial, -2 is a zero. $h(x)= x^3+8x^2+14x+4$. Express $h(x)$ as a product of linear factors.
Can someone please explain and help me solve?
| Okay, the first thing to do is polynomial division (or synthetic division, whichever you prefer).
Since -2 is a 0, we know that $(x+2)$ is a factor of $h(x)$. We then divide $h(x)$ by $(x+2)$
Dividing: $$\frac{x^3 + 8x^2 + 14x + 4}{x+2} = x^2 + 6x + 2$$
So $h(x)$ becomes: $$(x+2)*(x^2+6x+2)$$
Now you must use the Quadr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/852355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What's the intuition behind the 2D rotation matrix? Can anyone offer an intuitive proof of why the 2D rotation matrix works?
http://en.wikipedia.org/wiki/Rotation_matrix
I've tried to derive it using polar coordinates to no avail.
| There's two ways to think of it. At this stage, I think seeing both might be helpful.
Hard way: Let $(x,y) \in \Bbb R^2$ be represented by polar coordinates $(r, \varphi)$. I mean the relations $x = r \cos \varphi, y = r \sin \varphi$. So, let $(x_\theta, y_\theta)$ be the point after a rotation of $\theta$. Clearly, w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Basic induction proof methods so we're looking to prove $P(n)$ that
$$1^2+2^3+\cdots+n^3 = (n(n+1)/2)^2$$
I know the basis step for $p(1)$ holds.
We're going to assume $P(k)$
$$1^3+2^3+\cdots+k^3=(k(k+1)/2)^2$$
And we're looking to prove $P(k+1)$
What I've discerned from the internet is that I should be looking to add... | An example proof:
Let $P(k)$ denote the statement that $\sum\limits_{i = 1}^k i^3 = \left( \frac{k(k + 1)}{2} \right)^2$.
We wish to prove that for all $k \in \mathbb Z$ such that $k > 0$, $P(k)$.
We will prove this by induction on $k$.
In the base case, $k = 1$, and we have that $1^3 = 1 = \left(\frac{1(1 + 1)}{2}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/852689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Let $N$ and $M$ be two digit numbers. Then the digits of $M^2$ are those of $N^2$, but reversed.
Let $N$ be a two digit number and let $M$ be the number formed from $M$ by reversing $N$'s digits. The digits of $M^2$ are precisely those of $N^2$, but reversed.
$Proof$:
Since $N$ is a two digit number, we can write $N... | however there exists an infinite sequence of such numbers: we have
$12^2=144$
$102^2=10404$
$1002^2=1004004$
$10002^2=100040004$
$\dots\ \dots\ \dots$
and the same with digits reversed:
$21^2=441$
$201^2=40401$
$2001^2=4004001$
$20001^2=400040001$
$\dots\ \dots\ \dots$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/858138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Solve $a$ and $b$ for centre of mass in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ Given ellipse:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
What length do $a$ and $b$ have to be so the centre of mass is $S(4;2)$?
I've tried steps to solve the equation to $$y=b\sqrt{1-\frac{x^2}{a^2}}$$ and integrate
$$A=b\int_0^a{\sqrt{1-\frac{x... |
We can also use Pappus' (Second) Centroid Theorem, which states that the volume of a solid of revolution is equal to the area of the region revolved about the axis of symmetry times the circumference of the circular path "swept out" by the centroid of the region,
$$ V \ = \ A \ \cdot \ 2 \pi \ \overline{r} \ \ . $$
T... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Trigonometric Identities and formulas There are so many identities like $\sin2θ$, $\cos2θ$, $\tan2θ$, $\sin(θ/2)$, $\cos(θ/2)$ and $\tan(θ/2)$. there are other formulas too like $\cos(α-β)$, $\sin(α-β)$ etc and yes the sum and product formulas of trigonometric function...
I am stuck in memorizing all. Is there any simp... | For the multiple angle stuff it suffices to remember De Moivre's formula:
$$
(\cos x + i \sin x)^n = \cos (nx) + i \sin (nx).\,
$$
With it you easily get
$$
\begin{align}(\cos x + i \sin x)^2 &= \cos^2 x + 2i\sin x \cos x - \sin^2 x = (\cos^2 x - \sin^2 x) + i(2 \sin x \cos x)\\ &= \cos(2x) + i \sin (2x)\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/858756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$13\mid4^{2n+1}+3^{n+2}$ How can I prove that $4^{2n+1}+3^{n+2}$ is always divisible by 13?
| $$4^{2n+1}+3^{n+2}=16^n\cdot 4+3^n\cdot 9\\16^n\cdot4+3^n\cdot 9\equiv3^n\cdot4+3^n\cdot 9\pmod {13}\\3^n(4+9)\equiv3^n\cdot13\equiv0\pmod{13}$$
This can also be solved with induction,for $n=0$
$$4+3^2=13$$
Assume it holds for $n=k$
$$4^{2k+1}+3^{k+2}$$
Prove it holds for $n=k+1$
$$4^{2k+3}+3^{k+3}=16\cdot4^{2k+1}+3\cd... | {
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"source": "stackexchange",
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How to Show the following converges to $e^{\frac{t^2}{2}}$ How to prove that $$\lim_{n\to\infty}\left[\left[e^{t\sqrt{\frac{1-p}{np\vphantom{()}}}}-1-t\cdot\sqrt{\frac{1-p}{np}}-\frac{1}{2}t^2\left(\frac{1-p}{np}\right)\right]\cdot p \\+\left[e^{-t\sqrt{\frac{p}{n(1-p)}}}-1+t\cdot\sqrt{\frac{p}{n{(1-p)}}}-\frac{1}{2}\c... | Reacll that: $e^{x}=1+x+\frac{x^{2}}{2}+o(x^{2})$ for small $x$ which corresponds to large $n$. Taking $x=t\sqrt{\frac{1-p}{np}}$ and $x=-t\sqrt{\frac{p}{n(1-p)}}$ we see that the first two terms are $o(x^{2})$. Notice also that as $n\to\infty$ we have $(1+\frac{\frac{t^{2}}{2}}{n})^{n}\to e^{\frac{t^{2}}{2}}$. So we h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/859795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Generalization of Bernoulli's Inequality Is it possible to generalize Bernoulli's Inequality to $(x+y)^n \geq x + ny$ provided $x+y \geq 0 $ and $x \geq 1$ and $n$ is a positive natural number?
I was thinking that the proof follows by induction: In base case, have $n=1$. Then we have that $(x+y)^n = (x+y)^1 = x+y \ge... | There is one step that is not (always) valid. The inequality
$$x^2 + (k+1)xy \geqslant \frac{1}{x}(x^2+(k+1)xy)$$
only holds for $x+(k+1)y \geqslant 0$, if $x > 1$. If $x+(k+1)y < 0$ and $x > 1$, you have a strict inequality in the other direction,
$$x^2 + (k+1)xy < \frac{1}{x}(x^2+(k+1)xy).$$
However, in that case, th... | {
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"source": "stackexchange",
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Let $p$ be a prime of the form $3k+2$ that divides $a^2+ab+b^2$ for some integers $a,b$. Prove that $a,b$ are both divisible by $p$. Let $p$ be a prime of the form $3k+2$ that divides $a^2+ab+b^2$ for some integers $a,b$. Prove that $a,b$ are both divisible by $p$.
My attempt:
$p\mid a^2+ab+b^2 \implies p\mid (a-b)(a... | Use quadratic reciprocity.
$$4(a^2+ab+b^2)=(2a+b)^2+3b^2$$
so if
$$a^2+ab+b^2\equiv 0 \pmod p$$ then
$$(2a+b)^2\equiv -3b^2 \pmod p$$ and $-3$ is a quadratic residue so
$$\left(\frac{-3}{p}\right)=1.$$
However by reciprocity,
$$\left(\frac{-3}{p}\right)=\left(\frac{p}{-3}\right)=\left(\frac{2}{3}\right)=-1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/867413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Surface area of sphere $x^2 + y^2 + z^2 = a^2$ cut by cylinder $x^2 + y^2 = ay$, $a>0$ The cylinder is given by the equation $x^2 + (y-\frac{a}{2})^2 = (\frac{a}{2})^2$.
The region of the cylinder is given by the limits $0 \le \theta \le \pi$, $0 \le r \le a\sin \theta$ in polar coordinates.
We need to only calculate t... | I know what you r doing rong, I solved this a week ago, the same way you did.
You forgot that $\sqrt{\sin^2\theta} = |\sin\theta|$, not $\sin\theta$
$ $
$ $
This is how you might have done
$$ 2\int_{-\pi/2}^{\pi/2}\int_0^{a\cos\theta}\dfrac{a}{\sqrt{a^2-r^2}}\cdot r \cdot drd\theta$$
$ $
$$= 2\int_{-\pi/2}^{\pi/2}\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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modulo group defined by an algebraic relation I am asked if $\{n, n^{2}, n^{3}\}$ forms a group under multiplication modulo $m$ where $m = n + n^{2} + n^{3}.$
As an example we see that $\{2, 4, 8\}$ does form a group modulo $14,$ with identity $8,$ but am stuck starting the proof for the general case. Thanks in advance... | We work modulo $m=n+n^2+n^3$. Since $m|n^4-n$, we have that $n^4=n$. This implies* $n^3=1$, so $n^3$ is the identity. Also, when $x,y\in\{n,n^2,n^3\}=G$, we have $xy\in G$, because $xy=n^an^b=n^{a+b}=n^c$, where $c-1=a+b-1\mod 3$. Take for example $x=n^2$ and $y=n^3$. Then, $xy=n^5=n^4\cdot n=n\cdot n=n^2$. So in the e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/869268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Rational number trivial case Let $a,b,c$ denote rational numbers, such that $(a+b\sqrt[3]2+c\sqrt[3]4)^3$ is also rational. Prove that at least two of the numbers $a,b,c$ must be zero. Actually I confused of the beginning steps for this proof. Should I apply the strategies of contradictory proofs
| Since
$$(a+bx+cx^2)^3\equiv (6bc^2+3a^2c+3ab^2)x^2+(6ac^2+6b^2c+3a^2b)x+(4c^3+2b^3+a^3+12abc)\pmod{(x^3-2)}$$
if $(a+b\sqrt[3]{2}+c\sqrt[3]{2})^3$ belongs to $\mathbb{Q}$ then
$$(2bc^2+a^2c+ab^2)=0, \qquad (2ac^2+2b^2c+a^2b)=0,\tag{1}$$
hence:
$$ a^2 b^2 c^2 = (a^2c+ab^2)(ac^2+b^2 c) = a^3 c^3 + 2a^2 b^2 c^2 + ab^4 c,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/870990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Evaluate: $\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$ Find the following limit $$\lim_{x \to \infty} \,\, \sqrt[3]{x^3-1} - x - 2$$
How do I find this limit? If I had to guess I'd say it converges to $-2$ but the usual things like L'Hôpital or clever factorisation don't seem to work in this case.
| Take aside the $-2$, for the moment. You can use the identity
$$
a^3-b^3=(a-b)(a^2+ab+b^2),
$$
with $a=\sqrt[3]{x^3-1}$ and $b=x$. Then
\begin{align}
\lim_{x\to\infty}(\sqrt[3]{x^3-1}-x)&=
\lim_{x\to\infty}
\frac{
(\sqrt[3]{x^3-1}-x)
(\sqrt[3]{(x^3-1)^2}+x\sqrt[3]{x^3-1}+x^2)
}{\sqrt[3]{(x^3-1)^2}+x\sqrt[3]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 4
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maximum area of a rectangle inscribed in a semi - circle with radius r.
A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum.
My Try:
Let length of the side be $x$,
Then the length of th... |
Eqn of circle: x^2 + y^2 = r^2
|
area of rectangle = a = 2xy
==> a^2 = 4 x^2 y^2
==> a^2 = 4 x^2 (r^2 - x^2)
==> a^2 = 4 (x^2 r^2 - x^4)
|
differentiating:
==> 2a da/dt = 4 (2x r^2 - 4x^3)
since, da/dt=0
==> 2 x r^2 = 4 x^3
==> x = r/sqrt(2)
|
second derivative test:
==>4 (2 r^2 - 12 x^2)
==>8 r^2 - 24 r^2
==>(-16) r^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 3
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Find the remainder of the polynomial division $p(x)/(x^2-1)$ for some $p$ Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5. $
Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$.
I'm not sure how to do this, as the only way I know of dividin... | Plug in $1$ and $-1$ to get two values of $r(x)$ which is linear. From there you can get what $a,b$ are in $ax+b.$
Since
$$f(x)=g(x)(x+1)(x-1)+r(x)$$
we have
$$ f(1)=g(1)(1+1)(1-1)+r(1)=r(1)=-10$$
$$ f(-1)=g(1)(-1+1)(-1-1)+r(-1)=r(-1)=16$$
We know the remainder is of degree $1$, so
$r(x)=ax+b$
and now we know,
$$r(1)... | {
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"source": "stackexchange",
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Cyclic Group Presentation Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24.
This presentation was obtained using the Todd-Coxeter process for a subgroup of index 2 in the group presented in problem 476854.
| $$x^2=y^2x^2y$$
$$x=x^{-1}y^2x^2y=y^{-1}(yx^{-1}y^2)x^2y$$
$$x=y^{-1}x^9y$$
then we can say that;
$$y^2x^2y=x^2=y^{-1}x^{18}y$$
$$y^2x^2=y^{-1}x^{18}$$
$$y^3=x^{16}$$
Now, $$(xy^2)^2=yx^2$$
$$xy^2xy^2=yx^2$$
$$xy^2x(y^3)=yx^2y=y^3x^2y^2=x^{18}y^2$$
$$y^2x^{17}=x^{17}y^2$$
$$y^5x=xy^{5}$$
As a last step;
$$x^2=y^2x^2y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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A transcendental number from the diophantine equation $x+2y+3z=n$ Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation
$$x+2y+3z=n$$
Prove that
$$
\sum_{n=0}^{\infty} \frac{1}{D_{2n+1}}
$$
is a transcendental number.
| We have that
$$
\displaystyle D_{n} = \left \lfloor \frac {(n+3)^2}{12} \right \rceil
$$
where $\lfloor x \rceil$ is the nearest integer to $x$. You can find a detailed proof here (first pdf link, p.50).
It gives
$$
D_{2n+1} = \left \lfloor \frac {(n+2)^2}{3} \right \rceil.
$$
As Jack D'Aurizio explained, we deduce th... | {
"language": "en",
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"source": "stackexchange",
"question_score": "5",
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Trigonometric Identities help How do you solve this? I can't figure out what I should do.
$$\sin ^4\left(A\right)+\cos ^2\left(A\right)=\cos ^4\left(A\right)+\sin ^2\left(A\right)$$
Also, why is this equal zero? Can someone explain how that simplifies to be zero?
$$\frac{\left(\frac{1}{\cos \left(x\right)}\right)-1}{... | Note that $\sin^4 A-\cos^4 A=(\sin^2 A-\cos^2 A)(\sin^2 A+\cos^2 A)=\sin^2 A-\cos^2 A$.
For the other question, we have $\frac{1}{\cos x}-1=\frac{1-\cos x}{\cos x}$ and $\frac{1}{\cos x}+1=\frac{1+\cos x}{\cos x}$. When we divide, the $\cos x$ cancel, and we get $\frac{1-\cos x}{1+\cos x}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the maximun value of the expression $P=\sum \sqrt[3]{\frac{a^{2}+a}{a^{2}+a+1}}$ Let $a,b,c$ be positive real numbers such that $abc\leq 1$ .Find the maximun value of the expression
$P=\sqrt[3]{\frac{a^{2}+a}{a^{2}+a+1}}+\sqrt[3]{\frac{b^{2}+b}{b^{2}+b+1}}+\sqrt[3]{\frac{c^{2}+c}{c^{2}+c+1}}$
| Holder's inequality or Power Means will give
$$ \sum_{cyc} \frac{a^2+a}{a^2+a+1} \ge \frac19\left(\sum_{cyc} \sqrt[3]{\frac{a^2+a}{a^2+a+1}} \right)^3$$
with equality iff $a=b=c$. We claim that the LHS reaches a maximum of $2$ exactly when $a=b=c=1$, and show the same below.
This is equivalent to showing
$$ \sum_{cyc}... | {
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"source": "stackexchange",
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Proving inequality $3^{n^2} > (n!)^4$ Prove that $3^{n^2} > (n!)^4$ for all positive integers $n$.
I tried to use induction on this problem but failed to do so. I instead tried to prove $3^{2n+1}>(n+1)^4$, but couldn't come up with the solution.
| The induction-step should be pretty easy:
*
*Assume $\displaystyle3^{n^2}>(n!)^4$
*Prove $\displaystyle3^{(n+1)^2}>(n+1)!^4$:
*
*$\displaystyle3^{(n+1)^2}=3^{n^2+2n+1}$
*$\displaystyle3^{n^2+2n+1}=3^{n^2}3^{2n+1}$
*$\displaystyle3^{n^2}3^{2n+1}>(n!)^43^{2n+1}$
*$\displaystyle(n!)^43^{2n+1}=\frac{(n+1)!^4}{(n+... | {
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"timestamp": "2023-03-29T00:00:00",
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if $\frac{1}{(1-x^4)(1-x^3)(1-x^2)}=\sum_{n=0}^{\infty}a_{n}x^n$,find $a_{n}$ Let
$$\dfrac{1}{(1-x^4)(1-x^3)(1-x^2)}=\sum_{n=0}^{\infty}a_{n}x^n$$
Find the closed form $$a_{n}$$
since
$$(1-x^4)(1-x^3)(1-x^2)=(1-x)^3(1+x+x^2+x^3)(1+x+x^2)(1+x)$$
so
$$\dfrac{1}{(1-x^4)(1-x^3)(1-x^2)}=\dfrac{1}{(1-x)^3(1+x+x^2+x^3)(1+x+... | Note that $\left(1-x^2\right)\left(1-x^3\right)\left(1-x^4\right)$ divides $\left(1-x^{12}\right)^3$ and $\displaystyle\left(1-z\right)^{-3}=\sum_{n=0}^{\infty}{n+2 \choose 2}z^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/875792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
solving integral with complex analysis I have problems with understanding of the evaluation of this integral below. It has been a long a time ago since I had complex analysis.
where $a = (1-\sqrt y )^2$ and $b = (1+\sqrt y )^2$.
Now my question are: How do I substitute in the last 2 steps, how can I calculate the resi... | Note that
$$
\sqrt{(b-x)(x-a)}=\sqrt{\left(\frac{b-a}{2}\right)^2-\left(x-\frac{b+a}{2}\right)^2}\tag{1}
$$
Therefore, we can substitute
$$
x=\frac{b+a}{2}-\frac{b-a}{2}\cos(\theta)\quad\text{so that}\quad\sqrt{(b-x)(x-a)}=\frac{b-a}{2}\sin(\theta)\tag{2}
$$
Furthermore, by the definition of $y$,
$$
\frac{b+a}2=1+y\qqu... | {
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"url": "https://math.stackexchange.com/questions/875960",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Intuitive ways to get formula of cubic sum Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$
I think the cubic sum is squaring the arithmetic sum $$1^3 + 2^3 + \dots + n^3 = (1 ... | Here's a proof by induction
I came up with
about two years ago.
Let
$s_k(n)
=\sum_{i=1}^n i^k
$.
Want to show that
$s_3(n) = (s_1(n))^2
$.
If you want to
prove by induction
that $a(n) = b(n)$,
there are two possibilities:
(1) Show
$a(0)-b(0) = 0$
and
$a(n)-b(n) = 0
\implies
a(n+1)-b(n+1) = 0$;
(2) show
$a(0) = b(0)$
an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/876922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 4,
"answer_id": 3
} |
taylor series of $\ln(1+x)$? Compute the taylor series of $\ln(1+x)$
I've first computed derivatives (up to the 4th) of ln(1+x)
$f^{'}(x)$ = $\frac{1}{1+x}$
$f^{''}(x) = \frac{-1}{(1+x)^2}$
$f^{'''}(x) = \frac{2}{(1+x)^3}$
$f^{''''}(x) = \frac{-6}{(1+x)^4}$
Therefore the series:
$\ln(1+x) = f(a) + \frac{1}{1+a}\frac{... | You got the general expansion about $x=a$. Here we are intended to take $a=0$. That is, we are finding the Maclaurin series of $\ln(1+x)$.
That will simplify your expression considerably. Note also that $\frac{(n-1)!}{n!}=\frac{1}{n}$.
The approach in the suggested solution also works. We note that
$$\frac{1}{1+t}=1-t+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/878374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "56",
"answer_count": 4,
"answer_id": 0
} |
A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$? The following integrals are classic, initiated by L. Euler.
\begin{align}
\displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} \zeta(3)+\frac{93}{128} \zeta(5),
\\ \int_{0}^{\pi/2} x^2 \ln\cos x\:\... | Other series expansions can be found by noticing that:
$$\int\limits_{0}^{\frac{\pi}{2}} \frac{\log{\left(\cos{\left(x \right)} \right)}}{x}\, dx = -\frac{\pi}{2}\int\limits_{0}^{1}\int\limits_{0}^{1} \tan{\left(\frac{\pi x y}{2} \right)}\, dx\, dy \tag{1}$$
which enables us to use the series expansion of the tangent f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/879958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 4,
"answer_id": 3
} |
Summation of Infinite Geometric Series Determine the sum of the following series:
$$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} $$
My work:
$$\sum_{n=1}^{\infty } \frac{(-3)^{n-1}}{7^{n}} = \sum_{n=1}^{\infty } \frac{-1}{7} (\frac{3}{7})^{n-1}$$
$$\sum_{n=1}^{\infty } ar^{n-1} = \frac{a}{1-r} = \frac{\frac{-1}{7}}{1-... | $$-\frac{1}{3} \sum_{n=1}^{\infty} \frac{3^n}{7^n}=-\frac{1}{3} \sum_{n=1}^{\infty} \left ( \frac{3}{7} \right )^n=-\frac{1}{3} \sum_{n=0}^{\infty} \left ( \frac{3}{7} \right )^n+\frac{1}{3}=-\frac{1}{3} \frac{1}{1-\frac{3}{7}}+\frac{1}{3} \\ =-\frac{1}{3}\frac{7}{7-3}+\frac{1}{3}=-\frac{7}{12}+\frac{4}{12}=\frac{-1}{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/880019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
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