Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Prove line connecting intersection of tangents and opposite vertex bisects segment containing intersection of tangents and a vertex Let $\triangle ABC$ be an isosceles triangle with $AB=BC$. Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let the tangents at $A$ and $B$ intersect at $D$, and let $DC\cap\Gamma=E\ne... |
Let $|BA|=|BC|=a$, $|AC|=b$,
$\angle ABC=\beta$,
$\angle CAB=\angle ACB=\alpha=\tfrac\pi2-\tfrac\beta2$.
Since $\triangle AEC \cong\triangle FED$,
let's find the scaling factor,
\begin{align}
\frac{|EC|}{|DE|}
&=
\frac{|DC|-|DE|}{|DE|}
=
\frac{|DC|}{|DE|}-1.
\end{align}
From $\triangle DBC$:
\begin{align}
|DC|^2
&=
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1380515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Help with C is Euler's constant and $\Gamma(0)=\infty$ in paper I am referring to a paper by S. Nadarajah & S. Kotz.
The notation is simple enough to understand, however i having trouble with $C$ is Euler’s constant and $\Gamma(0)=\infty$
by equation (2.3) I have
$$F(z)= \displaystyle\lambda\int_{0}^{\infty}y^{-1-1}ex... | Look at this part of the formula given in Lemma 1:
$$
\begin{multline}
-\frac{2cp^{\alpha+1}}{\sqrt{\pi}}\Gamma(-\alpha-1)G\left(\frac{1}{2},\frac{3}{2},\frac{3+\alpha}{2},1+\frac{\alpha}{2}; -\frac{c^2p^2}{4}\right) \\
+ \frac{1}{c^\alpha\sqrt{\pi}\alpha}\Gamma\left(\frac{\alpha+1}{2}\right) G\left(-\frac{\alpha}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1383138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find min & max of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$ Problem: Find the maximum and minimal value of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$.
Since $x^2 > x$ (edit $x^2 \geq x$) for all $x \in \mathbb{R}$, $f$ is bowl-ish with a minimal value in the bottom.
This is a critical point which means t... | Your method to get the minimum gets in the entire plane, but you want the minimum on the unit circle $x^2+y^2=1$.
You can use Lagrange multipliers. You could also parameterize the unit circle on one variable then find the extrema on that variable. Two parametrizations are
$$x=\cos\theta,\quad y=\sin\theta,\quad 0\le\th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 0
} |
10th derivative of a function I want to find $f^{(10)}(0)$ where $f(x)=\ln(2+x^2)$.
I know that it can be done "by hand", but I believe there is a smarter way.
I think I should use Taylor series and the fact that $f^{(n)}(0)=a_n*n!$ , but I'm not sure how.
| \begin{align}
f(x) &= \ln(2 + x^{2}) = \ln 2 + \ln\left( 1 + \frac{x^{2}}{2}\right) \\
&= \ln 2 + \sum_{k=1}^{\infty} \frac{(-1)^{k-1} \, x^{2k}}{2^{k} \, k} \\
&= \ln 2 + \frac{x^{2}}{2} - \frac{x^{4}}{8} + \frac{x^{6}}{24} - \frac{x^{8}}{64} + \frac{x^{10}}{160} - \cdots \\
f(x) &= \ln 2 + \frac{x^{2}}{2!} - 3 \, \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1384164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
} |
Why $\int \frac{dx}{\sqrt{9-x^2}} = sin^{-1}\frac{x}{3}$? I don't understand how the two formulas are equal since the right side involves trigonometric sine that the left is devoid of.
| You can use the following substitution
$$ (*) \qquad x=\dfrac{3}{2}\sin t \implies \text{ d}x = \dfrac{3}{2} \cos t \text{ d}t $$
This'll yield the following equality
$$ \underbrace{\int \dfrac{1}{\sqrt{9-4x^2}} \text{ d}x \ \overset{(*)}= \ \dfrac{1}{3} \int \dfrac{1}{|\cos t|} \dfrac{3}{2} \cos t \text{ d}t}_{\becaus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1385533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
how to solve $3 - \cfrac{2}{3 - \cfrac {2}{3 - \cfrac {2}{3 - \cfrac {2}{...}}}}$ $$A = 3 - \cfrac{2}{3 - \cfrac {2}{3 - \cfrac {2}{3 - \cfrac {2}{...}}}}$$
My answer is:
$$\begin{align}
&A = 3 - \frac {2}{A}\\
\implies &\frac {A^2-3A+2}{A}=0\\
\implies &A^2-3A+2=0\\
\implies &(A-1)\cdot(A-2)=0\\
\implies &A=1\;\text{ ... | This continued fraction is the limit of
the sequence $a_n=3-2/a_{n-1}$. Computing the first few terms shows that $2$ is the correct limit; if our initial term $a_1=3$ were different then the limit could be $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1386966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
Solve this trigonometric equation $ \sin2x-\sqrt3\cos2x=2$ Solve equation: $$ \sin2x-\sqrt3\cos2x=2$$
I tried dividing both sides with $\cos2x$ but then I win $\frac{2}{\cos2x}$.
| $\bf{My\; Solution::}$ Given $\displaystyle \sin 2x -\sqrt{3}\cos 2x = 2$
We can write it as $$\displaystyle \sin 2x \cdot \frac{1}{2}-\cos 2x\cdot \frac{\sqrt{3}}{2} = 1\Rightarrow \sin \left(2x-\frac{\pi}{3}\right) =1=\sin \frac{\pi}{2}$$
Above we have used the formula $$ \sin \alpha\cdot \cos \beta-\cos \alpha\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1390718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 1
} |
Proving $\frac{1}{1\cdot3} + \frac{1}{2\cdot4} + \cdots + \frac{1}{n\cdot(n+2)} = \frac{3}{4} - \frac{(2n+3)}{2(n+1)(n+2)}$ by induction for $n\geq 1$ I'm having an issue solving this problem using induction. If possible, could someone add in a very brief explanation of how they did it so it's easier for me to understa... | Hint:
$$\frac { 1 }{ 2 } \left( \frac { 1 }{ n } -\frac { 1 }{ n+2 } \right) =\frac { 1 }{ n\left( n+2 \right) } $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
} |
$\frac{\pi}2 < \sum_0^\infty \frac{1}{1+n^2} < \frac{3\pi}4 $ Prove that:
$\frac{\pi}2 < \sum_0^\infty \frac{1}{1+n^2} < \frac{3\pi}4 $
What I've tried:
I solved the improper integral: $\int_0^\infty \frac{1}{1+x^2} = \lim_{b\to \infty} \arctan b -\arctan 0 = \frac{\pi}2 $.
Now, everything in the sum (and in the int... | The key result to use is that if $f(x)$ is a continuous strictly decreasing function, then $f(n+1) < \int_{n}^{n+1} f(x)\,dx < f(n)$.
If we take the sum over all values of $n$ starting at $n=0$ using the right inequality, we get that $$\frac{\pi}{2} = \int_{0}^\infty \frac{dx}{1+x^2} < \sum_{n=0}^\infty \frac{1}{1+n^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1391252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
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If $x^3+y^3=72$ and $xy=8$ then find the value of $x-y$. I recently came across a question,
If $x^3+y^3=72$ and $xy=8$ then find the value of $(x-y)$.
By trial and error I found that $x=4$ and $y=2$ satisfies both the conditions. But in general how can I solve it analytically? I tried using $a^3+b^3=(a+b)(a^2+b^2-ab)... | You may solve this by brute force. Write $x=8/y$. The first equation now reads
$x^3+\frac{512}{x^3}=72$.
Multiply both sides by $x^3$, and you have
$x^6+512-72x^3=0$.
Define $u\equiv x^3$, and the equation becomes
$u^2-72u+512=0$.
Use the quadratic formula to solve for $u$, solve back for x, and solve for y.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
} |
Finding $\int \sec^3x\,dx$ I've tried substitution, replacing $\sec^2x$ with $\tan^2x + 1$, and parts and I just hit dead ends every time... Do you need knowledge of higher-level calculus to solve this?
| $\bf{My\; Solution::}$ Let $\displaystyle I = \int \sec^3 x dx = \int \frac{1}{\cos^3 x} dx = \int\frac{1}{\sin^3\left(\frac{\pi}{2}-x\right)}dx$
Now Let $\displaystyle \left(\frac{\pi}{2}-x\right) = t\;,$ Then $dx = -dt$
So Integral $\displaystyle I = -\int\frac{1}{\sin^3 t} dt =-\frac{1}{8}\int \frac{1}{\sin^3 \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1392749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 5
} |
Triangle Area problem I've been trying to solve the following:
Let $ABC$ be a triangle with sides $a, b $ and $ c$, inradius $r$ and exradii $r_a, r_b$ and $r_c$. If $A'B'C'$ is another triangle with sides $\sqrt{a}, \sqrt{b}$ and $\sqrt{c}$ show that $Area(A'B'C')=\frac {\sqrt{r(r_a+r_b+r_c)}} {2}$.
I tried to combin... | Brute force method:
Firstly, Heron's formula can be simplified in this form:
The area $\Delta$ of a triangle with sides $a,b,c$ is given by $16\Delta^2 $ $= (a + b + c)(b + c - a) (c + a - b)(a + b - c) \\= 2(a^2 b^2 + b^2 c^2 + c^2 a^2) - (a^4 + b^4 + c^4)$
Therefore area of triangle with sides as $\sqrt a, \sqrt b, \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1393074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why is this sum zero? I have been looking at the following sum (for any positive integer $n$)
$$\left(1-\frac{1^2}{n}\right) + \left(1-\frac{1}{n}\right)\left(1-\frac{2^2}{n}\right) + \left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)\left(1-\frac{3^2}{n}\right) + \ldots $$
Note that the $i$th term in the sum has $i$... |
The following holds true for $n\geq 1$
\begin{align*}
\sum_{k=1}^{\infty}\left(1-\frac{k^2}{n}\right)\prod_{j=1}^{k-1}\left(1-\frac{j}{n}\right)=0\tag{1}
\end{align*}
Note, the empty product is set equal to $1$. We start by transforming the product into a somewhat more convenient form .
\begin{align*}
\sum_{k=1}^{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1394580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 2
} |
Proof of the identity: $c\sin \frac{A-B}{2} \equiv (a-b) \cos \frac{C}{2}$ Trigs is not my strongest apparently...
I need to prove $c\sin \frac{A-B}{2} = (a-b) \cos \frac{C}{2}$ for a general triangle $ABC$.
Here is what I do, or rather, here is how I fail at proving it:
$\cos \frac{C}{2} \equiv \sin \frac{A+B}{2}$, s... | Here's a trigonograph:
(This space intentionally left blank.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1394674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to prove the trigonometric identity $\frac{\cot x}{1- \tan x} + \frac{\tan x}{1 - \cot x} - 1 = \sec x \csc x$ I am doing some practice questions for a Math class and I was told that similar questions would be in the exam. So I need to learn this but I have no idea where to even start with this question:
$$\frac{\c... | Given $$\displaystyle \frac{\cot x}{1-\tan x}+\frac{\tan x}{1-\cot x} = \frac{1}{\sin x}\cdot \left(\frac{\cos^2 x}{\cos x-\sin x}\right) - \frac{1}{\cos x}\cdot \left(\frac{\sin^2 x}{\cos x-\sin x}\right)$$
So $$\displaystyle = \frac{\cos^3 x-\sin^3 x}{\sin x\cdot \cos x\cdot (\cos x-\sin x)} = \frac{(\cos x-\sin x)\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Family of Lines Problem:
Consider a family of straight lines $(x+y)+\lambda(2x-y+1)=0$. Find the equation of the straight line belonging to this family which is farthest from $(1,-3)$.
$$$$
Any help with this problem would be really appreciated!
| We can write the equation as $\displaystyle (2\lambda+1)x+(1-\lambda)y+\lambda+1 =0$
Now Perpendicular Distance of Line from Point $\bf{P(1,-3)}$ is $ = PQ=\displaystyle \left|\frac{2\lambda+1+3\lambda-3+\lambda+1}{\sqrt{(2\lambda+1)^2+(1-\lambda)^2}}\right|$
Now $\displaystyle PQ = \left|\frac{6\lambda-1}{\sqrt{5\lamb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1395626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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find the limit of $ \lim_{(x,y) \rightarrow (0,0)}\frac{2xy^3+x^2y^3}{x^4+2y^4}$ find the limit of $$ \lim_{(x,y) \rightarrow (0,0)}\frac{2xy^3+x^2y^3}{x^4+2y^4}$$
I have absolutely no idea how to proceed with that. I would prefer a solution that would involve use of squeeze theorem
| Let $y=kx$. So,
$$
\frac{2xy^3+x^2y^3}{x^4 + y^4} = \frac{k^3x^3(2x+x^2)}{x^4(1+k^4)} = \frac{k^3}{1+k^4}(x+2)
$$
and limit depends on $k$; so, it doesn't exists.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Finding the derivative of an absolute value This one I just don't know how to derive.
$\ln\|x^4\cos x\|$
I know the derivative of $\ln\ x$, is just $\frac{1}{x}$ . It is the absolute value that throws me off. My question is, does the absolute value stay as is or does it disappear?
*
*$\frac{1}{|x^4\cos x|}$
*$\frac... | HINT:
$$\frac{d|x|}{dx}=\text{sgn}(x)$$
for $x\ne0$. Now, use the chain rule, but make sure that $x^4 \cos x\ne0$
SPOILER ALERT: SCROLL OVER SHADED AREA TO REVEAL ANSWER
Let $y=x^4\cos x$. Then, we have $$\frac{d}{dx}\log|x^4\cos x|=\frac{d\log |y|}{d|y|}\text{sgn(y)}\frac{dy}{dx}=\frac{1}{|x^4\cos x|}\text{sgn}(x^4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
$\int \frac{dx}{\sin x \sqrt{\sin(2x+\alpha)}}$ $\int \frac{dx}{\sin x \sqrt{\sin(2x+\alpha)}}$
I tried:
$\int \frac{dx}{\sin x \sqrt{\sin(2x+\alpha)}}=\int \frac{dx}{\sin x \sqrt{\sin2x\cos \alpha+\cos 2x\sin \alpha}}$,then i could not solve and changed the method.
Let $\sqrt{\sin(2x+\alpha)}=u\Rightarrow dx=\frac{u d... | Given $\displaystyle \int\frac{1}{\sin x\sqrt{\sin (2x+a)}}dx = \int\frac{1}{\sin x\sqrt{\sin 2x\cdot \cos a+\cos 2x\cdot \sin a}}dx$
So we get $\displaystyle = \int\frac{1}{\sin x\sqrt{2\sin x\cdot \cos x\cdot \cos a+(\cos^2 x-\sin^2 x)\cdot \sin a}}$
$\displaystyle = \int\frac{1}{\sin^2 x\sqrt{2\cot x\cdot \cos a+(\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1396772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Show that either $1+\alpha+\alpha^2+.....+\alpha^{p-1}=0$ or $1+\alpha+\alpha^2+.....+\alpha^{q-1}=0$,but not both together. Let a complex number $\alpha,\alpha\neq1$,be a root of the equation $z^{p+q}-z^p-z^q+1=0$,where $p$ and $q$ are distinct primes.Show that either $1+\alpha+\alpha^2+.....+\alpha^{p-1}=0$ or $1+\al... | If $a$ is a root of $1+z+z^2+\cdots+z^{m-1}=0\ \ \ \ (1),a\ne1$
and is a root of $\dfrac{z^m-1}{z-1}=0$
If $a$ is a root of $1+z+z^2+\cdots+z^{n-1}=0\ \ \ \ (2),a$ is a root of is a root of $\dfrac{z^n-1}{z-1}=0$
Now Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$ says $(a^m-1,a^n-1)=a^{(m,n)}-1$
$$\impli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1397796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Find inverse of 15 modulo 88. Here the question: Find an inverse $a$ for $15$ modulo $88$ so that $0 \le a \le 87$; that is, find an integer $a \in \{0, 1, ..., 87\}$ so that $15a \equiv1$ (mod 88).
Here is my attempt to answer:
Find using the Euclidean Algorithm, we need to find $\gcd(88, 15)$, that must equal to $1$ ... | For a systemetic way, you have to use the Extended Euclidean Algorithm: if you have a Bézout's relation:
$$u\times 88+v\times 15=1$$
the you know the inverse of $15$ is $v\bmod88$.
The following shows how to display the computation, taking into account that, in Euclid's algorithm, the remainder at each step is a linear... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1399743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Evaluation of $\sum_{r=1}^{n}r\cdot (r-1)\cdot \binom{n}{r} = $ Evaluation of $\mathop{\displaystyle \sum_{r=1}^{n}r\cdot (r-1)\cdot \binom{n}{r} = }$
$\bf{My\; Try::}$ Given $$\displaystyle \sum_{r=1}^{n}r\cdot (r-1)\cdot \binom{n}{r}\;,$$ Now Using the formula $$\displaystyle \binom{n}{r} = \frac{n}{r}\cdot \binom{n-... | $\sum_{r=0}^{n} r(r-1)\binom{n}{r} = 2 \sum_{r=0}^n \binom{n}{r} \binom{r}{2}$. The summand is the number of ways of choosing from $n$ objects an $r$-tuple and a 2-subtuple, so the right-hand sum is just the number of ways of choosing a 2-tuple multiplied by the number of possible tuples the 2-subtuple could have been ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find $\lim\limits_{x\to0}\frac{\sqrt{1+\tan x}-\sqrt{1+x}}{\sin^2x}$ Find:
$$\lim\limits_{x\to0}\frac{\sqrt{1+\tan x}-\sqrt{1+x}}{\sin^2x}$$
I used L'Hospital's rule, but after second application it is still not possible to determine the limit. When applying Taylor series, I get wrong result ($\frac{-1}{6}$). What met... | $$\lim_{x \to 0} \dfrac{\sqrt{1+\tan x}-\sqrt{1+x}}{\sin^2x}=L$$
Using L'hopital:
$$\lim_{x \to 0} \dfrac{\dfrac{\sec^2x}{2\sqrt{1+\tan x}}-\dfrac{1}{2\sqrt{1+x}}}{2\sin x \cos x}=L$$
Reordering the denominator:
$$\lim_{x \to 0} \dfrac{\dfrac{\sec^2x}{2\sqrt{1+\tan x}}-\dfrac{1}{2\sqrt{1+x}}}{\sin 2x }=L$$
Using L'hop... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1400657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent?
Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent?
Y will always depend on X . NO... | What you want to show is that
$$
P(X=x \wedge Y=y) = P(X=x)P(Y=y)
$$
for all values $x$ and $y$ of $X$ and $Y$.
Let $Z$ be the number of tosses between the 5th and 6th head. Then
\begin{align*}
P(X=x \wedge Y=y)
&= \sum_{z=1}^\infty \underbrace{\binom{x-1}{4}p^5q^{x-5}}_{X=x} \underbrace{\vphantom{\binom{x-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of trignometric limit I want to find the following limit without L'Hospital.
$ \lim_{x \to \frac{3π}{4}} \frac{1+(\tan x)^{\frac13}}{1-2(\cos x)^2}$
Maybe I should try to get rid of the radical.
| let us make a change of variable $x = 3\pi/4 + h.$ then we have
$$\begin{align}\tan(x) &= \tan(3\pi/4 + h) = \frac{-1 + \tan h}{1 + \tan h} \\
&= (\sin h - \cos h)(\sin h + \cos h \cdots)^{-1} \\
&=(-1 + h \cdots)(1+h +\cdots)^{-1} \\
&=(-1 + h \cdots)(1-h +\cdots) \\
&=-1+2h+\cdots\\
\tan^{1/3} x &=-1 + \frac23 h+\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
If $\frac{dy}{dx}\frac{dx}{dy} = 1$, does $\frac{d^2 y}{dx^2} \frac{d^2 x}{dy^2} = 1$? I know $\frac{dy}{dx}\frac{dx}{dy} = 1$ because the chain rule says $1 = \frac{dy}{dy} = \frac{dy}{dx}\frac{dx}{dy}$. But does $\frac{d^2 y}{dx^2} \frac{d^2 x}{dy^2} = 1$? Or would that be too good to be true?
| If you want an example where the derivatives don't vanish, consider $y=x^3$ (so $x=y^{1/3}$). Then:
$$
\frac{dy}{dx}=3x^2
$$
and
$$
\frac{dx}{dy}=\frac{1}{3}y^{-2/3}=\frac{1}{3}x^{-2}
$$
and
$$
\frac{dy}{dx}\frac{dx}{dy}=3x^2\cdot\frac{1}{3x^2}=1.
$$
On the other hand,
$$
\frac{d^2y}{dx^2}=6x
$$
and
$$
\frac{d^2x}{dy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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Arithmetic Derivatives: Arithmetic Logarithmic Derivative Problem In Calculus, whenever we see a constant and want to take the derivative of it, it always is 0. However in Number Theory, we have something called the arithmetic derivative in which we can differentiate to get some nonzero term.
So we can denote the arit... | Since $a=b$ is a trivial solution. For $a \neq b$:
$$L(a)=L(b) \\ \frac{a'}{a}=\frac{b'}{b} \\ a'b-b'a=0 \\ \frac{a'b-b'a}{b^2}=0 \\ \left(\frac{a}{b}\right)'=0$$
Now what is $\displaystyle\left(\frac{a}{b}\right)'$?
Lemma 1:
Let prime factorization of $n$ be $\prod_{i=1}^{k} p_i^{a_i}$, where $p_i$'s are distinct p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403375",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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} |
Inequality between altitude and sides in triangle Let $a,b,c$ be the side lengths and $h_a,h_b,h_c$ the altitudes each connect a vertex to the opposite side and are perpendicular to that side. Then we need to prove $h_a^2+h_b^2+h_c^2\leq\dfrac14(a+b+c)^2$.
I know the inequality $h_a^2+h_b^2+h_c^2\leq\dfrac34(a^2+b^2+c... | Following ASCII advocate's idea, the problem boils down to proving that:
$$ 4\Delta^2\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\leq (a+b+c)^2 \tag{1}$$
or, using Heron's formula,
$$ (a+b-c)(a-b+c)(-a+b+c)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\leq (a+b+c). \tag{2}$$
Through Ravi substitution ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Solve $\cos \frac{4x}{3}=\cos x+1$
Solve the equation \begin{equation} \cos \frac{4x}{3}=\cos x+1\tag 1\end{equation}
I had tried by taking $\cos\dfrac x3=t$ and from this we have $\displaystyle\cos\frac{4x}3=2\left(2t^2-1\right)^2-1; \cos x=4t^3-3t$
$(1) \iff t\left(8t^3-4t^2-8t+3\right)=0$
But I can't solve $8t^3-4... | There is a trigonometric method for solving cubic equations. Not very pleasant but doable. Applying the method to the case of $$8t^3-4t^2-8t+3=0$$ Using $A=-\frac{1}{2}$, $B=-1$, $C=\frac{3}{8}$, $Q=-\frac{13}{36}$, $R= -\frac{43}{432}$, $D=-\frac{257}{6912}$, $$\theta= \cos ^{-1}\left(-\frac{43}{26 \sqrt{13}}\right)$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Discrete mathematics question $$(n+1)^2+(n+2)^2+(n+3)^2+\dots+(2n)^2=\frac{n(2n+1)(7n+1)}{6}$$
Prove the statement using mathematical induction.
| Let $s(n)=1^2+2^2+...+n^2=\sum_{j=1}^nj^2.$. You want calculate $s(2n)-s(n)$. We know that $s(n)=\frac{n(n+1)(2n+1)}{6}$, then $s(2n)=\frac{2n(2n+1)(4n+1)}{6}$. It follows that
$$s(2n)-s(n)=\frac{2n+1}{6}(8n^2+2n-n^2-n)=\frac{2n+1}{6}(7n^2+n)=\frac{n(2n+1)(7n+1)}{6}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
If $x^4+7x^2y^2+9y^4=24xy^3$,show that $\frac{dy}{dx}=\frac{y}{x}$ If $x^4+7x^2y^2+9y^4=24xy^3$,show that $\frac{dy}{dx}=\frac{y}{x}$
I tried to solve it.But i got stuck after some steps.
$x^4+7x^2y^2+9y^4=24xy^3$
$4x^3+7x^2.2y\frac{dy}{dx}+7y^2.2x+36y^3.\frac{dy}{dx}=24x.3y^2\frac{dy}{dx}+24y^3$
$\dfrac{dy}{dx}=\dfrac... | Notice, we have $$x^4+7x^2y^2+9y^4=24xy^3\tag 1$$
Now, differentiating both the sides w.r.t. $x$ as follows $$\frac{d}{dx}(x^4+7x^2y^2+9y^4)=\frac{d}{dx}(24xy^3)$$
$$4x^3+14x^2y\frac{dy}{dx}+14xy^2+36y^3\frac{dy}{dx}=72xy^2\frac{dy}{dx}+24y^3$$
$$\frac{dy}{dx}(14x^2y+36y^3-72xy^2)=24y^3-4x^3-14xy^2$$
$$\frac{dy}{dx}=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Correctness of the definite integral Consider the integral
\begin{eqnarray*}
I & = & \int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}(1+\sqrt{1-x^{2}})}\\
& = & \int_{-1}^{0}\frac{dx}{\sqrt{1-x^{2}}(1+\sqrt{1-x^{2}})}+\int_{0}^{1}\frac{dx}{\sqrt{1-x^{2}}(1+\sqrt{1-x^{2}})}
\end{eqnarray*}
Use the substitutions $u=-\sqrt{1-x^{2}}... | Let $$\displaystyle I = \int_{-1}^{1}\frac{dx}{\sqrt{1-x^{2}}(1+\sqrt{1-x^{2}})}dx= \int_{0}^{1}\frac{dx}{\sqrt{1-x^{2}}(1+\sqrt{1-x^{2}})}dx$$
Above we Used $\displaystyle \bullet \int_{-a}^{a}f(x)dx = 2\int_{0}^{a}f(x)dx\;,$ If $f(-x) = f(x)$
Now Put $x=\sin \phi\;,$ Then $dx = \cos\phi d\phi$ and Changing limit, We ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Understanding how to use $\epsilon-\delta$ definition of a limit I finally understand the geometric intuition behind the $\epsilon-\delta$ definition of a limit, which is actually quite neat:
But I'm having trouble actually using the definition to come to a conclusion.
For (a solved) example, to prove that $\lim_{n\ri... | HINT:
Note that
$$\left|a_n-\frac{1}{3}\right|=\left|\frac{2n+5}{6n-3}-\frac{1}{3}\cdot\frac{2n-1}{2n-1}\right|=\left|\frac{2n+5-(2n-1)}{6n-3}\right|=\left|\frac{6}{6n-3}\right|=\left|\frac{2}{2n-1}\right|$$
Taking $n_0\ge\max\left(\frac{1}{\epsilon}+1,2\right)$ we have the inequality, since
$$n\ge n_0\quad\Longrightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1407328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) :
Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear systems:
First ... | *
*Gaussian elimination :
(n+1)(n-1)+(n)(n-2)+...+(3)(1)=sigma(k=1 to n-1)(k+2)(k)=n^3/3+n^2/2-5n/6
2.Backward gaussian method :
2(n-1)+2(n-2)+...+2(1)=2*sigma(k=1 to n-1)(k)=n^2-n
3.Find the solution :
n
Sum of multi./div. = sum of 1,2,3 = n^3/3+3n^2/2-5n/6
I tried to upload my picture to explain it in detail bu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1407732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
How to solve $ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $ I am trying to solve this integral, I think that it could be solve using the complex. $$ \int \limits _0 ^{2\pi} \dfrac{dx}{(\alpha +\beta\cos x)^2} $$
| Let $$\displaystyle I = \int \frac{1}{(\alpha+\beta \cos x)^2}dx\;,$$ Now Let $$\displaystyle t = \frac{\beta+\alpha\cos x}{\alpha+\beta \cos x}$$
So $$\displaystyle \frac{dt}{dx} = \frac{\left(\alpha+\beta \cos x\right)\cdot -(\alpha \sin x)-(\beta+\alpha \cos x)\cdot (-\beta \sin x)}{(\alpha+\beta\cos x)^2} = \frac{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
cyclic three variable inequality
Let $a,b,c$ be nonnegative real numbers and $a+b+c=3$. Prove the inequality
$$
\sqrt{24a^2b+25}+\sqrt{24b^2c+25}+\sqrt{24c^2a+25}\le 21
$$
I have tried to find the solution using classical inequalities, but failed. Any idea?
| By C-S
$$\left(\sum_{cyc}\sqrt{24a^2b+25}\right)^2\leq\sum_{cyc}(24a^2b+25)(a+3b+5c)\sum_{cyc}\frac{1}{a+3b+5c}$$
Thus, it remains to prove that
$$\sum_{cyc}(24a^2b+25)(a+3b+5c)\sum_{cyc}\frac{1}{a+3b+5c}\leq441$$ or
$$\sum_{cyc}(648a^2b+25(a+b+c)^3)(a+3b+5c)\sum_{cyc}\frac{1}{a+3b+5c}\leq441(a+b+c)^3$$ or
$$\sum_{cyc}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1413271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
The differential equation $\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;$ I am learning differential equations and can do the basic examples. However, how can you solve the differential equation
$$\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;?$$
| You already have nice solutions with substitutions. If one cannot come up with the right substitution, maybe this is a solution that works better:
Multiplying the differential equation with $y$ gives
$$
yy'=\frac{y^2}{x}-1,
$$
or
$$
\frac{1}{2}\bigl(y^2\bigr)'=\frac{y^2}{x}-1.
$$
This is a linear and first order differ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1414760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Bound on maximum angle between vectors I have two vectors $\mathbf v_1$ and $\mathbf v_2$:
$$\mathbf v_1 = \begin{pmatrix}x_1\\y_1\\z_1\end{pmatrix},
\mathbf v_2 = \begin{pmatrix}x_2\\y_2\\z_2\end{pmatrix}$$
The components of these vectors each differ by at most a constant $\epsilon$:
\begin{align}
|x_2 - x_1| &\le \e... | Assume that $\mathbf{v}_1$ is fixed and $\mathbf{v}_2$ can be varied within the constraints. Furthermore, assume none of $x_1,y_1,z_1$ are zero, and $\epsilon \ll \min(|x_1|,|y_1|,|z_1|)$.
WLOG consider only $\mathbf{v}_1$ in the positive octant, i.e. $x_1,y_1,z_1>0$ and with $x_1\ge y_1\ge z_1$. The angle between $\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1416857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Trying to show $|\overrightarrow{a}\times\overrightarrow{b}|^2=|\overrightarrow{a}|^2|\overrightarrow{b}|^2-(\overrightarrow{a}⋅\overrightarrow{b})^2$
If $\overrightarrow{a} = \langle a_1, a_2, a_3 \rangle$ and $\overrightarrow{b} = \langle b_1, b_2, b_3 \rangle$, then the cross product of $\overrightarrow{a}$ and $\o... | Let me try.
You already have: $$\begin{eqnarray}LHS &=& a_1^2b_2^2 + a_1^2b_3^2 + a_2^2b_1^2 + a_2^2b_3^2 + a_3^2b_1^2 + a_3^2b_2^2 - 2 a_1a_2b_1b_2 - 2 a_2a_3b_2b_3 - 2a_3a_1b_3b_1\\ & =& a_1^2b_2^2 + a_1^2b_3^2 + a_2^2b_1^2 + a_2^2b_3^2 + a_3^2b_1^2 + a_3^2b_2^2 + a_1^2b_1^2 + a_2^2b_2^2 + a3^2b_3^2 -(a_1^2b_1^2 + a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1418422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Minimum value of $\cos x+\cos y+\cos(x-y)$ What is the minimum value of $$ \cos x+\cos y+\cos(x-y). $$ Here $x,y$ are arbitrary real numbers. Mathematica gives (with NMinimize) $-3/2$. But I don't know if this is correct and if so, how to prove it.
| The local max or min of a 2-variable function comes where both partial derivatives are 0. So if we say that
$$
z = \cos x+\cos y+\cos(x-y)
$$
then
$$
\frac{\partial z}{\partial x} = \sin x + \sin(x-y) = 0
$$
and
$$
\frac{\partial z}{\partial y} = \sin y - \sin(x-y) = 0
$$
adding the two yields
$$
\begin{align}
\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1419019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Confusing probability problems based on product rule and combinations I am going thru probability exercise. Faced first problem:
Book Q1. Ten tickets are numbered 1,2,3,...,10. Six tickets are selected at random one at a time with replacement. What is the probability the largest number appearing on the selected ticket... | Addressing your Q1, the difference between the answer you gave and the answer that the book gives is that your solution assumes that a specific selection is 7. That is, $\frac{7^5}{10^6}$ is the probability that a fixed ticket is 7 and the other five take values from 1 to 7 while the books solution just assumes that a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1420232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Solving the infinite series $1-\frac{2^3}{1!}+\frac{3^3}{2!}-\frac{4^3}{3!}+\cdots$ I have the following question:
Evaluate the infinite series:
$$S=1-\frac{2^3}{1!}+\frac{3^3}{2!}-\frac{4^3}{3!}+\cdots$$
(a) $\displaystyle\frac1e$ (b) $\displaystyle\frac{-1}e$ (c) $\displaystyle\frac{2}e$ (d) $\displaystyle\fr... | Since $(n+1)^3 = \color{red}{1}\cdot n(n-1)(n-2)+\color{red}{6}\cdot n(n-1) +\color{red}{7}\cdot n+\color{red}{1}$ we have:
$$\begin{eqnarray*}\sum_{n=0}^{+\infty}\frac{(-1)^n (n+1)^3}{n!} &=& \frac{13}{2}+\sum_{n\geq 3}\frac{(-1)^n (n+1)^3}{n!}\\&=&\frac{13}{2}+\color{red}{1}\cdot\sum_{n\geq 3}\frac{(-1)^n}{(n-3)!}+\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1423107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Find the area of the circle that falls between the circle $x^2+y^2=5$ and the lines $x^2-4y^2+6x+9=0$
Find the area of the circle that falls between the circle $x^2+y^2=5$ and the lines $x^2-4y^2+6x+9=0$.
I tried to solve this question. The lines are $x-2y+3=0$ and $x+2y+3=0$ which intersect at $(-3,0)$ and the circl... | If I may refer to sketch of Omran Kouba, you need even not rotate each pink segment by $\pm \tan ^{-1} \frac12$, you can straightaway use the standard formula of area of circular segment ( subtract sector and triangle areas)..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1425476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Evaluating the limit $\lim_{x\rightarrow 1} \frac{\sqrt{x+3}-2}{x-1}$ The question is :
$$\lim_{x\rightarrow 1} \frac{\sqrt{x+3}-2}{x-1}$$
I know I probably have to do some sort of factorisation of the numerator in order to cancel the denominator, but the surd has me stumped I'm afraid.
| If you multiply both the numerator and the denominator by the conjugate of the expression, you'll get:
\begin{align*}
\lim_{x \to 1}\frac{\sqrt{x + 3} -2}{x - 1} \cdot \frac{\sqrt{x+3}+2}{\sqrt{x+3}+2} & = \lim_{x \to 1} \frac{x + 3 - 4}{(x-1)(\sqrt{x + 3} + 2)}\\
& = \lim_{x \to 1} \frac{x-1}{(x-1)(\sqrt{x + 3} + 2)}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Find roots of equation $(x^2+1)\cdot \arccos\left(\frac{2x}{1+x^2}\right)+2x\cdot \mathrm{sgn}(x^2-1)=0$
Find roots of equation $(x^2+1)\cdot \arccos\left(\frac{2x}{1+x^2}\right)+2x\cdot \mathrm{sgn}(x^2-1)=0$
One root is $x=1$ (checking functions $\arccos$ and $\mathrm{sgn}$).
Second root is $x=0.442$.
How to find t... | We can start by simplifying the sign function. We know that $x^2 - 1$ is negative between $-1 < x < 1$ and positive or zero otherwise (I'm assuming that's the definition you're using for it), so we have the following two equations:
$$(x^2 + 1) \cdot \arccos{\left(\frac{2x}{1 + x^2}\right)} - 2x = 0, -1 < x < 1$$
$$(x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1426602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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If $(x+1)^4+(x+3)^4=4$ then how to find the sum of non-real solutions of the given equation? If $(x+1)^4+(x+3)^4=4$ then how to find the sum of non-real solutions of the given equation?
I took $x+2=t$ and got $t^2=-6+\sqrt(40)/2$.How to proceed?
| Put $t=x+2$ then you obtain $$(t-1)^4 +(t+1)^4 =4 .$$
hence $$((t-1)^2)^2 -2(t^2-1 )^2 +((t+1)^2)^2 =4 -2(t^2-1 )^2$$ hence $$16t^2 =4 -2(t^2-1 )^2$$ and let $s=t^2 -1 $ we obtain $$16 s +12 +2s^2 =0$$ therefore $$s^2 +8s +6 =0$$ thus $$x=-2\pm i\sqrt{3+\sqrt{10}}$$ or $$x=-2 \pm \sqrt{\sqrt{10}- 3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1428093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding the exact value of $\tan(\cos^{-1} x)$
Find the exact values of:
a) $\tan(\cos^{-1} x)$
b) $\cos(\tan^{-1} x)$
c) $\sec(\sin^{-1} x)$
My work is the following.
a) $\tan x = \sin x / \cos x$ so
$$\tan(\cos^{-1} x) = \sin(\cos^{-1} x) / \cos(\cos^{-1} x) = \sin(\cos^{-1}x) / x$$
b) $\cos(\tan^{-1} x... | Useful things to note
$$
\tan^2 x + 1 = \sec^2 x \implies \tan x = \pm\sqrt{\frac{1}{\cos^2 x}-1}
$$
And
$$
\cos^2 x+\sin^2 x = 1\implies \cos x =\pm\sqrt{1-\sin^2 x}
$$
Can you use these to your advantage?
| {
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"url": "https://math.stackexchange.com/questions/1430536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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What is the sum $\sum_{k=0}^{n}k^2\binom{n}{k}$? What should be the strategy to find
$$\sum_{k=0}^{n}k^2\binom{n}{k}$$
Can this be done by making a series of $x$ and integrating?
| Let $$\displaystyle S=\sum^{n}_{k=0}k^2 \binom{n}{k} \;,$$ Now using $\displaystyle \bullet \; \binom{n}{k} = \frac{n}{k}\cdot \binom{n-1}{k-1}$
So we get $$\displaystyle S = \sum^{n}_{k=0}k^2 \cdot \frac{n}{k}\binom{n-1}{k-1}\;,$$ Again using $\displaystyle \bullet \; \binom{n-1}{k-1} = \frac{n-1}{k-1}\cdot \binom{n-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1431112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Solving an integral with substitution method I want a hint for the following integral:
$$\int \frac{dx}{x^{2}\sqrt{1- a^{2}+x^{2}}}$$
where $a$ is a real constant.
In fact,
$$\int \frac{dx}{x^{2}\sqrt{1- a^{2}+x^{2}}} = -\frac{\sqrt{1- a^{2}+x^{2}}}{x(1- a^{2})}$$
why?
| Assume a$\ne $$1 or-1$,Put $x$=$\sqrt {1-a^2}$$tan$$\theta$ and then simplify..
$\int \frac{dx}{x^{2}\sqrt{1- a^{2}+x^{2}}}$=$\int \frac{sec^2\theta d\theta}{({1-a^2})tan^2\theta sec\theta}$=$\int \frac{cos\theta d\theta}{({1-a^2})sin^2\theta}$=
$\int \frac{d(sin\theta)}{({1-a^2})sin^2\theta}$=$\frac{-1}{({1-a^2})sin\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1431824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Minimum of $\frac{1}{x+y+z}+\frac{1}{x+y+w}+\frac{1}{x+z+w}-\frac{2}{x+y+z+w}$ Let $x,y,z,w\geq 0$ and $0\leq x+y,y+z,z+x,x+w,y+w,z+w\leq 1$. What is the minimum of $$F(x,y,z,w)=\frac{1}{x+y+z}+\frac{1}{x+y+w}+\frac{1}{x+z+w}-\frac{2}{x+y+z+w}?$$
We have $F(1/2,1/2,1/2,1/2)=F(1,0,0,0)=1$, so the minimum is at most $1$.... | Not a very easy problem, but the following solution is nice enough ;)
Denote $a=y+z,b=z+w,c=y+w$ then $y=\frac{a+c-b}{2},z=\frac{a+b-c}{2},w=\frac{b+c-a}{2}$.
The constraints become
\begin{align}
x\ge 0 \\
a+b-c\ge 0 \\
b+c-a\ge 0\\
c+a-b\ge 0 \\
0\le a,b,c \le 1 \\
x+ \frac{a+b-c}{2} \le 1 \\
x+ \frac{b+c-a}{2} \le 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Coefficient of the generating function $G(z)=\frac{1}{1-z-z^2-z^3-z^4}$ I am seeking the coefficient $a_n$ of the generating function
$$G(z)=\sum_{k\geq 0} a_k z^k = \frac{1}{1-z-z^2-z^3-z^4}$$
The combinatorial background of this question is to solve the recurrence
$$a_n=a_{n-1}+a_{n-2}+a_{n-3}+a_{n-4},\qquad (a_0,a_... | Here is another variant to extract the coefficients $a_n$ resulting in a different formula.
We obtain
\begin{align*}
\frac{1}{1-z-z^2-z^3-z^4}&=\frac{1}{1-z(1+z+z^2+z^3)}\\
&=\frac{1}{1-z\frac{1-z^4}{1-z}}\tag{1}\\
&=\frac{1-z}{1-2z+z^5}\\
&=(1-z)\sum_{k=0}^\infty z^k(2-z^4)^k\tag{2}\\
&=(1-z)\sum_{k=0}^\infty z^k\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1434941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Determinant of remainder of a primitive matrix modulo 2 I'm trying to prove the following relation for a matrix $A\in \mathbb{Z}^{m\times m} $, $m\geq 2$. It is assumed that the characteristic polynomial of $A$ is primitive modulo $2$:
If $C$ is defined to be the remainder of $A^{2^m-1}\pmod{4}$ , i.e.
$$A^{2^m-1}\e... | The claim is not true for $m=2$: Take
$$
A=\pmatrix{1& 3\\1& 0}, \quad p_A = x^2-x-3
$$
$$
A^3 =\pmatrix{7 & 12 \\ 4 & 3} \equiv \pmatrix{3&0\\0&3} \equiv I + 2I \pmod 4
$$
hence $2C \equiv 2I \pmod 4$, $C\equiv I \pmod 2$, and $C+I\equiv 0\pmod2$, in contradiction to the claim $det(C+I)\equiv 1\pmod 2$.
Here is anoth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1435233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Does the series $\sum\frac{(-1)^n cos(3n)}{n^2+n}$ converge absolutely? $\sum|\frac{(-1)^n \cos(3n)}{n^2+n}| \le \sum\frac{1}{n^2+n}$
since $-1 \le cos(3n) \le 1$
$\sum\frac{1}{n^2+n} = \sum\frac{1}{n(n+1)} = \sum(\frac{1}{n} - \frac{1}{n+1})$
$\int(\frac{1}{x} - \frac{1}{x+1}) dx = \log|x| + \log|x+1| + C$
which diver... | Of course the mistake lies in the integral test:
$$
\frac{d}{dx} \left( \log |x| + \log |x+1| \right) = \frac{1}{x}+\frac{1}{x+1},
$$
so your antiderivative is wrong.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find $n$, where its factorial is a product of factorials I need to solve $3! \cdot 5! \cdot 7! = n!$ for $n$.
I have tried simplifying as follows:
$$\begin{array}{}
3! \cdot 5 \cdot 4 \cdot 3! \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3! &= n! \\
(3!)^3 \cdot 5^2 \cdot 4^2 \cdot 7 \cdot 6 &= n! \\
6^3 \cdot 5^2 \cdot 4^2 \... | We have
$$3!\cdot 5!\cdot 7!=(1\cdot 2\cdot 3)\cdot (1 \cdot 2\cdot 3\cdot 4\cdot 5)\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7,$$
and combining some of those gives
$$1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot \underbrace{(2\cdot 4)}_8\cdot \underbrace{(3\cdot 3)}_9\cdot \underbrace{(2\cdot 5)}_{10}=10!$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1440482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 7,
"answer_id": 1
} |
Evaluate the limit of ratio of sums of sines (without L'Hopital): $\lim_{x\to0} \frac{\sin x+\sin3x+\sin5x}{\sin2x+\sin4x+\sin6x}$ Limit to evaluate:
$$\lim_{x \rightarrow 0} \cfrac{\sin{(x)}+\sin{(3x)}+\sin{(5x)}}{\sin{(2x)}+\sin{(4x)}+\sin{(6x)}}$$
Proposed solution:
$$
\cfrac{\sin(x)+\sin(3x)+\sin(5x)}{\sin(2x)+\sin... | HINT:
$$\sin (ax) =(ax)+O(x^3)$$
as $x\to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
$e^{\cot^2\theta}+\sin^2\theta-2\cos^22\theta+4=4\sin\theta$ in $[0,10\pi]$ Number of solution of the equation $e^{\cot^2\theta}+\sin^2\theta-2\cos^22\theta+4=4\sin\theta$ in $[0,10\pi]$ is $(A)2\hspace{1cm}(B)3\hspace{1cm}(C)4\hspace{1cm}(D)5$
In this question,$e^{\cot^2\theta}$ varies from $1$ to infinity,whereas $RH... | We rewrite the equation as $e^{\cot^2\theta}=2\cos^22\theta-\sin^2\theta+4\sin\theta-4$. The equation has a period of 2π. L.H.S.≥1, then R.H.S.≥1, i.e. $2\cos^22\theta-\sin^2\theta+4\sin\theta-4≥1$, then we have $\cos4\theta≥(sin\theta-2)^2$. Since $\cos4\theta$∈[-1,1] and $(sin\theta-2)^2$∈[1,9] for θ∈[0,2π], θ must b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to find this indefinite integral? Find $\displaystyle\int \frac{dx}{2\sqrt x+\sqrt{x+1}+1}$
I think I should change dx and let x = something. What is your suggestion?
| Let me try to use $x=\sinh^2\theta$, then $dx=2\sinh\theta\cosh\theta d\theta=\sinh2\theta d\theta$
$$\int \frac{dx}{2\sqrt x+\sqrt{x+1}+1}$$
$$=\int \frac{\sinh2\theta d\theta}{2\sinh\theta+\cosh\theta+1}$$
$$=\int \frac{(e^{2\theta}-e^{-2\theta})/2}{e^\theta+(e^\theta+e^{-\theta})/2+1}d\theta$$
$$=\int \frac{e^{4\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Deriving an equation for the Sum of Alternating Combinations Consider the equation $$A(n,m) = \sum_{i = 0}^m (-1)^i {n \choose i}$$
To get a feel for it, I let $n=5$ and $m \in \{1,2,...,5\}$
for $m=1$ $$A(5,1) = (-1)^0 {5 \choose 0} - {5 \choose 1} = 1 - 5 = -4$$
for $m=2$, $$A(5,2) = A(5,1) + {5 \choose 2} = -4 + 10 ... | Using the binomial identities: Negating the upper index $\binom{n}{i}=(-)^i\binom{i-n-1}{i}$ and parallel sumation $\sum_{k=0}^{n}\binom{m+k}{k}=\binom{n+m+1}{n}$ we get :
$$
\eqalign{
\sum_{i=0}^{m}(-)^i\binom{n}{i} &= \sum_{i=0}^{m}(-)^i(-)^i\binom{i-n-1}{i} \cr
&= \sum_{i=0}^{m}\binom{i-n-1}{i} = \binom{m-n}{m} \cr
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1442903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$[\cos x+\sin x]=[\cos x]+[\sin x]$,where [.] is the greatest integer function. Solve the equation in interval $[0,\pi]:[\cos x+\sin x]=[\cos x]+[\sin x]$,where [.] is the greatest integer function.
How should i start this question,breaking it into intervals is difficult.Please guide me.
| Here $x\in \left[0,\pi\right]\;,$ Then $0\leq \sin x\leq 1$. So we get $\displaystyle \lfloor \sin x\rfloor = 0,1$
Similarly $x\in \left[0,\pi\right]\;,$ Then $-1 \leq \cos x\leq 1$. So we get $\displaystyle \lfloor \cos x\rfloor = -1, 0,1$
Similarly $x\in \left[0,\pi\right]\;,$ Then $-1 \leq \sin x+ \cos x\leq \sqrt{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Integral involving cube root and seventh root Find the value of $$\int_{0}^{1} (1-x^7)^{\frac{1}{3}}-(1-x^3)^{\frac{1}{7}}\:dx$$
My Approach:
Let $$I_1=\int_{0}^{1} (1-x^7)^{\frac{1}{3}}dx$$ and
$$I_2=\int_{0}^{1} (1-x^3)^{\frac{1}{7}}dx$$
For $I_1$ substitute $x^7=1-t^3$ so $dx=\frac{-3t^2}{7}(1-t^3)^{\frac{-6}{7}}\:... | Consider the area in the first quadrant bounded by the curve $x^7+y^3 = 1$.
The curve in the first quadrant can be written as $y = (1-x^7)^{1/3}$ for $0 \le x \le 1$. Hence, the area bounded by the curve is given by $\displaystyle\int_{0}^{1}(1-x^7)^{1/3}\,dx$.
We can also write the curve in the first quadrant as $x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Cauchy–Schwarz type Inequality for 4 variables
For $x,y,z,t \geqslant 0$, prove that
$$ \tag{1} (x+2y+3z+4t)(x^2+y^2+z^2+t^2) \geqslant \frac{35-\sqrt{10}}{54}(x+y+z+t)^3$$
Observations
*
*This inequality is not symmetric nor cyclic. We cannot order the variables.
*This has 4 variables. My first attempt is to a... | This is a typical problem that can be solved by brute force with homogeneization techniques and Lagrange multipliers. Since the inequality is homogeneous, it is not restrictive to assume $x^2+y^2+z^2+t^2=1$, then compute
$$ \max_{x^2+y^2+z^2+t^2=1} f(x,y,z,t)=\max_{x^2+y^2+z^2+t^2=1}\frac{x+2y+3z+4t}{(x+y+z+t)^3} $$
by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1445853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
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Uniformly distributed variables: what does the sum reveal. Say $U_1, U_2 \sim U(1,0)$ are independent uniformly randomly distributed variables on $[0, 1]$.
What lower bound $C$ should I enforce on their sum in order to believe (with a probability $p$) that at least one of the variables themselves is equal to at least $... | Denote $A = \max \{ U_1, U_2\}$ and $B=U_1 + U_2$. And I suppose you want to find the smallest such $C$ for any $p$, thus the least information about $U_1 + U_2$.
First, suppose $C \le 0$, then we have $$\mathbb{P}(A \ge p \mid B \ge C) = \mathbb{P}(A \ge p) = 1- p^2.$$
If we want $\mathbb{P}(A \ge p)=p$, we must solve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Solving $\lim_{x \to 4}\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}$ without L'Hopital. I am trying to solve the limit
$$\lim_{x \to 4}\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}$$
Without using L'Hopital.
Evaluating yields $\frac{0}{0}$. When I am presented with roots, I usually do this:
$$\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}} \cdot \frac{3+\... | For every $x\in [-5,5]\setminus\{4\}$ we have:
\begin{eqnarray}
\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}&=&\frac{(3-\sqrt{5+x})\color{green}{(3+\sqrt{5+x})}\color{blue}{(1+\sqrt{5-x})}}{(1-\sqrt{5-x})\color{blue}{(1+\sqrt{5-x})}\color{green}{(3+\sqrt{5+x})}}=\frac{[9-(5+x)](1+\sqrt{5-x})}{[1-(5-x)](3+\sqrt{5+x})}\\
&=&\frac{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1447028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Prove $u^2_n-u_{n-1}u_{n+1}=(-1)^n$ if $u_1=1,u_2=2,u_n=u_{n-1}+u_{n-2},n>2$ Prove $u^2_n-u_{n-1}u_{n+1}=(-1)^n$ if $u_1=1,u_2=2,u_n=u_{n-1}+u_{n-2},n>2$
If $n=3$ equality is true.
How to prove this by induction?
| We may use the following famous trick: Notice that
$$ \begin{pmatrix}u_{n+2} \\ u_{n+1}\end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} u_{n+1} \\ u_n \end{pmatrix}. $$
Recursively using this relation, we have
$$ \begin{pmatrix}u_{n+1} & u_n \\ u_{n} & u_{n-1} \end{pmatrix} = \begin{pmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1448428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What are the methods of dividing numbers to get weird values like $16\over 17$ without a calculator? I tried estimating it to somewhere near $16\over 20$, but it's a far stretch from getting the actual $16\over 17$. How can one do so? Conventionally, I think for numbers such as $50\over 17$, or for any large numbers, w... | Long division:
$$
\begin{array}{cccccccccc}
& & & 0 & . & 9 & 4 & 1 & 1 & 7 & 6 \\ \\
17 & ) & 1 & 6 & . & 0 & 0 & 0 & 0 & 0 & 0 \\
& & 1 & 5 & & 3 \\ \\
& & & & & 7 & 0 \\
& & & & & 6 & 8 \\ \\
& & & & & & 2 & 0 \\
& & & & & & 1 & 7 \\ \\
& & & & & & & 3 & 0 \\
& & & & & & & 1 & 7 \\ \\
& & & & & & & 1 & 3 & 0 \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1449027",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 7,
"answer_id": 4
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Probability that the sum of 6 dice rolls is even Question:
6 unbiased dice are tossed together. What is the probability that the sum of all the dice is an even number?
I think the answer would be 50%, purely by intuition. However, not sure if this is correct. How should I go about solving such a problem?
| The generating function approach:
$$P(x)=(1+x+x^2+x^3+x^4+x^5)^6=\sum a_i x^i$$
Then $a_i$ counts the number of ways of getting a total of $i+6$ from $6$ dice.
Now, to find the even terms, you can compute $$\frac{P(1)+P(-1)}{2}=\sum_i a_{2i}.$$
But $P(1)=6^6$ and $P(-1)=0$. So $$\frac{P(1)+P(-1)}{2}=\frac{6^6}{2},$$ or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1451818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 8,
"answer_id": 5
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Evalutating $\lim_{x\to0}\frac{1-\cos x}{x^2}$ $$\lim_{x\to0}\frac{1-\cos x}{x^2}$$
I know there are many ways to calculate this. Like L'Hopital. But for learning purposes I am not supposed to do that. Instead, I decided to do it this way:
Consider that $\cos x = 1- \sin^2 \frac{x}{2}$ (from the doulbe-angle formulas h... | $$\begin{align}
\lim_{x\to 0}\frac{1-\cos x}{x^2}\frac{1+\cos x}{1+\cos x} & = \lim_{x\to 0}\frac{\sin^2x}{x^2(1+\cos x)}\\
& = \lim_{x\to 0}\frac{\sin^2x}{x^2}\lim_{x\to 0}\frac 1{1+\cos x}\\
& = 1\cdot\frac 12=\frac 12.
\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1452958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Correct understanding of Manly's $S$ statistic formula I am working on social associations of bats and have a problem with correct understanding of formula from:
Bejder, Lars, D Fletcher, and S BrÄger. 1998. “A Method for Testing
Association Patterns of Social Animals.” Animal Behaviour 56 (3):
719–25. doi:10.1006... | Hint: Let your matrix be M. Then
$$\begin{pmatrix} 1&1&1\end{pmatrix} \times M \times \begin{pmatrix} 1\\1\\1\end{pmatrix}=S$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1453290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to solve $\lim _{x\to \infty \:}\left(\sqrt{x+\sqrt{1+\sqrt{x}}}-\sqrt{x}\right)$ Solved I stuck in this limit, I tried to solve it and gets 1/2 as result. Yet, I was wrong because I forgot a square. Please need help!
$\lim _{x\to \infty \:}\left(\sqrt{x+\sqrt{1+\sqrt{x}}}-\sqrt{x}\right)$
Note: it's $+\infty$
Tha... | For clarity, I will use $y=\sqrt{x}$ The limit becomes
$$
\lim_{y\to \infty}( \sqrt{y^2+\sqrt{1+y}}-y)= \lim_{y\to \infty}( \sqrt{y^2+\sqrt{1+y}}-y)\frac{\sqrt{y^2+\sqrt{1+y}}+y}{\sqrt{y^2+\sqrt{1+y}}+y}
=\lim_{y\to \infty} \frac{\sqrt{1+y}}{\sqrt{y^2+\sqrt{1+y}}+y} \\
=\lim_{y\to \infty}\frac{\sqrt{y}\sqrt{1/y+1}}{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1455623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Where is this converging to on $y=x$? Well I was playing with graphs and I started plotting equations as the following:
$$\underbrace{x+y}_{degree=1}=1 \tag{1}$$
$$\underbrace{x^2+y^2+xy}_{degree=2}+\underbrace{x+y}_{degree=1}=1 \tag{2}$$
$$\underbrace{x^3+y^3+x^2y+xy^2}_{degree=3}+\underbrace{x^2+y^2+xy}_{degree=2}+\... | If you think of the both sides of that equation as functions of $x$, you're looking for a value $x$ with
$$
f(x) = g(x)
$$
where $f$ is a sum and $g$ is a constant function.
The sum happens to also be
$$
F'(x)
$$
where
\begin{align}
F(x)
&= \sum_{n=1}^\infty x^{n+1}\\
&= \sum_{n=2}^\infty x^{n}\\
&= -1-x + \sum_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1456527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How to simplify $\sin^4 x+\cos^4 x$ using trigonometrical identities? $\sin^{4}x+\cos^{4}x$
I should rewrite this expression into a new form to plot the function.
\begin{align}
& = (\sin^2x)(\sin^2x) - (\cos^2x)(\cos^2x) \\
& = (\sin^2x)^2 - (\cos^2x)^2 \\
& = (\sin^2x - \cos^2x)(\sin^2x + \cos^2x) \\
& = (\sin^2x - \c... | \begin{align}
\sin^4 x +\cos^4 x&=\sin^4 x +2\sin^2x\cos^2 x+\cos^4 x - 2\sin^2x\cos^2 x\\
&=(\sin^2x+\cos^2 x)^2-2\sin^2x\cos^2 x\\
&=1^2-\frac{1}{2}(2\sin x\cos x)^2\\
&=1-\frac{1}{2}\sin^2 (2x)\\
&=1-\frac{1}{2}\left(\frac{1-\cos 4x}{2}\right)\\
&=\frac{3}{4}+\frac{1}{4}\cos 4x
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1458305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Li Shanlan's combinatorial identities I am struggling to prove the following combinatorial identities:
$$(1)\quad\sum_{r=0}^m \binom{m}{r}\binom{n}{r}\binom{p+r}{m+n} = \binom{p}{m}\binom{p}{n},\quad \forall n\in\mathbb N,p\ge m,n$$
$$(2)\quad\sum_{r=0}^m \binom{m}{r}\binom{n}{r}\binom{p+m+n-r}{m+n} = \binom{p+m}{m}\bi... | Very striking identities. Hard to guess how Li Shanlan discovered them...
Concerning (1), multiply by $x^n y^m z^p$ and add up.
In the rhs we have
$$
\sum_{p\ge 0}z^p\sum_{n=0}^p {p \choose n} x^n \sum_{m=0}^p {p \choose m} y^m = \sum_{p\ge 0}(1+x)^p(1+y)^p z^p = \frac{1}{1-z(1+x)(1+y)}.
$$
The lhs is tougher. Denoti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1460712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Prove using factor theorem. Using factor theorem, show that $a+b$,$b+c$ and $c+a$ are factors of
$(a + b + c)^3$ - $(a^3 + b^3 + c^3)$
How do we go about solving this ?
Thanks in advance !
| To show, for example that $a+b$ is a factor of $p(a,b,c) = (a+b+c)^3 -(a^3 + b^3 + c^3)$, we consider $p$ as a polynomial in $a$, with constants $b$ and $c$, let's write $$ p_{b,c}(a) = p(a,b,c)$$
Now $a - (-b)$ is a factor of $p_{b,c}$ iff $-b$ is a root of $p_{b,c}$. We have
\begin{align*}
p_{b,c}(-b) &= (-b + b +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1460959",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Proving differentiability I just had a question on proving differentiability by showing that the difference quotient exists. I understand in the case of a function like $f(x)=x^2$, where you end up with $((x+h)^2 - x^2)/h = 2x + h = 2x$ as h goes to infinity, but in the case of a function such as $1/x^n$, how do you ad... | METHOD 1: BRUTE FORCE USE OF THE DEFINITION OF THE DERIVATIVE
We can actually approach this with brute force using the binomial expansion. We can write
$$(x+h)^n=x^n\left(1+n\frac hx+\frac{n(n-1)}{2!}\frac{h^2}{x^2}+\cdots+\frac{h^n}{x^n}\right)$$
Therefore,
$$\begin{align}
\frac{1}{(x+h)^n}-\frac{1}{x^n}&=\frac{1}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1462526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
What is the radius of the circle through $(-1,1)$ and touching the lines $x\pm y=2?$ What is the radius of the circle through $(-1,1)$ and touching the lines $x\pm y=2?$
The lines $x+y=2$ and $x-y=2$ are perpendicular to each other and the circle is touching both the lines,these lines are tangents to the circle.Let po... | Here is another approach. The general equation for a circle is
$${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {R^2}$$
with the center at $(a,b)$ and the radius $R$. As the line $x+y=2$ is tangent to your circle, hence the line and the circle just have one intersection that you called point $P$, so the coor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1465571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Find the equation of the locus of the intersection of the lines below Find the equation of the locus of the intersection of the lines below
$y=mx+\sqrt{m^2+2}$
$y=-\frac{ 1 }{ m }x+\sqrt{\frac{ 1 }{ m^2 +2}}$
By graphing, I have got an ellipse as locus : $x^2+\dfrac{y^2}{2}=1$.
The given lines form tangent and normal ... | Notice,
Solving the given equations of the straight lines: $y=mx+\sqrt{m^2+2}$ & $y=-\frac{1}{m}x+\sqrt{\frac{1}{m^2+2}}$,
as follows
$$mx+\sqrt{m^2+2}=-\frac{1}{m}x+\sqrt{\frac{1}{m^2+2}}$$
$$mx+\frac{x}{m}=\frac{1}{\sqrt{m^2+2}}-\sqrt{m^2+2}$$
$$\frac{m^2+1}{m}x=\frac{1-m^2-2}{\sqrt{m^2+2}}\implies x=\frac{-m}{\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Limit of $\frac{\frac{1}{e}(1+x)^{1/x}-1+\frac{x}{2}}{x^2}$ when $x\to0$
Find the limit of $\dfrac{\frac{1}{e}(1+x)^{1/x}-1+\frac{x}{2}}{x^2}$ when $x\to0$.
I tried applying L'Hospital rule, but it is not working here. How should I solve this?
| With the definition of $a^b$ we have for the numerator
\begin{align*}
(1+x)^\frac{1}{x} \cdot \exp(-1) -1+\frac{x}{2}&= \exp\left(\frac{1}{x} \cdot \ln(1+x) \right) \cdot \exp(-1) -1+\frac{x}{2}\\
&=1-\frac{x}{2}+\frac{11 x^2}{24}-1+\frac{x}{2} +\mathcal{O}(x^3)\\
&= \frac{11x^2}{24} +\mathcal{O}(x^3)
\end{align*}
Now... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
find the sum of $ \sum_{n=1}^{\infty} \frac{x^{2n}(-1)^n}{n} $ $ \sum_{n=1}^{\infty} \frac{x^{2n}(-1)^n}{n} $
compute the sum of this
$\frac{1}{1+x^2}$ = $ \sum_{n=1}^{\infty} x^{2n}(-1)^n $
and then integrate both sides
$ \sum_{n=1}^{\infty} \frac{x^{2n+1}(-1)^n}{2n+1} $
but how to get n from there
| You know that:
$\begin{align}
\frac{1}{1 - z}
&= \sum_{n \ge 0} z^n \\
\int_0^z \frac{\mathrm{d} t}{1 - t}
&= - \ln (1 - z) \\
&= \sum_{n \ge 0} \frac{z^{n + 1}}{n + 1}
\end{align}$
so that:
$\begin{align}
\sum_{n \ge 1} \frac{(-1)^n x^{2 n}}{n}
&= - \ln (1 + x^2)
\end{align}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1468740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
if $5\nmid a$ or $5\nmid b$, then $5\nmid a^2-2b^2$. I have a homework as follow:
if $5\nmid a$ or $5\nmid b$, then $5\nmid a^2-2b^2$.
Please help to prove it.
EDIT: MY ATTEMPT
Suppose that $5\mid a^2-2b^2$, then $a^2-2b^2=5n$,where $n\in Z$,
then $a^2-2b^2=(a+\sqrt2b)(a-\sqrt2b)=5n$,
Since 5 is a prime number, we ge... | Hint$\ \ (ab^{-1})^2\equiv 2\,\Rightarrow\,(ab^{-1})^4\equiv 4\,$ contra little Fermat
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1471722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How to find the absolute maximum of $f(x) = (\sin 2\theta)^2 (1+\cos 2\theta)$ for $0 \le \theta \le \frac{\pi}2$? How to find the absolute maximum of $f(x) = (\sin 2\theta)^2 (1+\cos 2\theta)$ for $0 \le \theta \le \frac{\pi}2$?
I found the derivative to be $f'(x)= (2\sin 4\theta)(1+\cos 2\theta) + (\sin 2\theta)^2... | In order to find the stationary points of $g^2$ it is enough to find the roots of $g$ and the stationary points of $g$. Since:
$$ \sin^2(2\theta)(1+\cos(2\theta)) = 2\sin^2(2\theta)\cos^2(\theta) $$
we just need to find the stationary points of:
$$ 2\sin(2\theta)\cos(\theta) = \sin(3\theta)+\sin(\theta) $$
that occur f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
On evaluating the Riemann zeta function, including that $\zeta(2)\gt \varphi$ where $\varphi$ is the golden ratio A week ago, I got the following :
For a positive integer $k$, using Cauchy–Schwarz inequality,
$$\left(\sum_{n=1}^{\infty}\frac{1}{n^k(n+1)^k}\right)^2\lt \left(\sum_{n=1}^{\infty}\frac{1}{n^{2k}}\righ... | Let's look at the
first few terms
of each side of
$$\zeta(2k)
\gt \dfrac{1+\sqrt{1+4\left(\sum_{n=1}^{\infty}\dfrac{1}{n^k(n+1)^k}\right)^2}}{2}.
$$
$\zeta(2k)
\approx 1+\frac1{4^k}+\frac1{9^k}
$
and
$\sum_{n=1}^{\infty}\dfrac{1}{n^k(n+1)^k}
\approx \frac1{2^k}+\frac1{6^k}
$
so,
since
$\sqrt{1+x}
\approx 1+\frac{x}{2}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1474931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
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How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$ How should I calculate the below limit
$$\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$$
I have no idea where to start from.
| First we use an observation by @Stan in the comment. Note that as $(1 +\frac{x}{n})^n$ is increasing in $n$ whenever $|x|<n$,
$$ \left(\frac{k}{n}\right)^n = \left(1 + \frac{k-n}{n}\right)^n \le e^{k-n}, $$
(here we assume that $x:= k-n$ is fixed and varies the remaining two $n$'s. This sequence is increasing and ten... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1475930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 1,
"answer_id": 0
} |
Calculate $\sum\limits_{n=1}^{\infty} \frac{a\cos(nx)}{a^2+n^2}$
I have to calculate $\sum\limits_{n=1}^{\infty} \frac{a\cos(nx)}{a^2+n^2}$ for $x\in(0,2\pi)$.
I have used the function $f(x)=e^{ax}$ and I have calculated the Fourier coefficients which are:
$$a_0=\dfrac 1{2a} \dfrac {e^{2\pi a}-1}{\pi}$$
$$a_n=\dfrac ... | Or equivallently, we can choose : $f(x)=\cosh ax$, then
$$f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nx + b_n \sin nx$$
where, since $f(x)$ is symmetric, $b_n =0$ and also
$$a_n=\frac{2}{\pi}\int_0^\pi\cosh ax \cos nx \;\mathrm{d}x=\frac{2a}{\pi}\frac{\sinh\pi a}{a^2+n^2}\,\cos{\pi n}$$
hence
$$\cosh ax= \frac{2}{\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1479666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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A trigonometry equation: $3 \sin^2 \theta + 5 \sin \theta \cos \theta - 2\cos^2 \theta = 0$
$$3 \sin^2 \theta + 5 \sin \theta \cos \theta - 2\cos^2 \theta = 0$$
What are the steps to solve this equation for $ \theta $?
Because, I am always unable to deal with the product $\sin \theta \cos \theta$.
| Factorise this into the quadratic:
$$3 \sin^2 \theta + 5 \sin \theta \cos \theta - 2\cos^2 \theta = 0$$
$$(3 \sin \theta - \cos \theta )(sin \theta + 2\cos \theta) = 0$$
This gives 2 equations.
$$3 \sin \theta - \cos \theta = 0$$
$$\sin \theta + 2\cos \theta = 0$$
You should be able to solve it by dividing both by cos.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1481232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
How to prove this inequality about $xyz=1$ Let $x,y,z>0$,and such $xyz=1$, show that
$$\dfrac{x+y}{x^3+x}+\dfrac{y+z}{y^3+y}+\dfrac{z+x}{z^3+z}\ge 3$$
I tried use $AM-GM$ inequality
$$\dfrac{x+y}{x^3+x}+\dfrac{y+z}{y^3+y}+\dfrac{z+x}{z^3+z}\ge 3\sqrt{\dfrac{(x+y)(y+z)(z+x)}{(x^3+x)(y^3+y)(z^3+z)}}$$
which shows that $... | Since $\frac{x+y}{x+x^3}=\frac{(x+y)yz}{(x+x^3)yz}=\frac{1+y^2z}{1+x^2}$ and similar equalities hold, we may rewrite the inequality as:
$$\sum \frac{1+y^2z}{1+x^2}\ge 3$$.
According to Cauchy-Schwarz:
$$\sum \frac{1+y^2z}{1+x^2}=\sum \frac{(1+y^2z)^2}{(1+x^2)(1+y^2z)}\ge \frac{(3+\sum y^2z)^2}{\sum (1+x^2)(1+y^2z)}$$
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1484203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 2,
"answer_id": 1
} |
Finding all positive integer solutions for $x+y=xyz-1$ How do I manually solve $x+y=xyz-1$ assuming that $x, y$ and $z$ are positive integers? I was able to guess all possible solutions, but I do not know how to show that these are the only ones:
$x=1, y=1, z=3$
$x=1, y=2, z=2$
$x=2, y=1, z=2$
$x=2, y=3, z=1$
$x=3, y=2... | Note that for $x,y$ positive integers $$xy\ge x+y+1$$ for $x>2$ and $y\ge 3$ (or vice verse). Therefore you can control only $y=1$(or vice verse) and $y=x=2$
If $x=1$ the equation becomes:
$$y(z-1)=1$$ and the solutions are $z=2$ $y=2$ and $z=3$, $y=1$. The same thing for $y=1$. While for $y=x=2$ there aren't solutio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1484330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
How to calculate the line integral and substitute $dx\,\ dy$ in a question on Green's theorem The question states that:
Verify Green's Theorem on the plane for $\oint_C (2x-y^3)dx-xydy$ where C is the boundary of the region enclosed by the circles $x^2+y^2=1$ and $x^2+y^2=9$.
My attempt:
*
*While verifying the L.H.... | On a circle the radius $r$ is a constant, so we have:
$$
x=r\cos \theta \quad \Rightarrow \quad dx=-r\sin \theta d\theta
$$
and
$$
y=r\sin \theta \quad \Rightarrow \quad dy=r \cos \theta d \theta
$$
(this is the mistake in your solution, I suppose).
So, for a circle of radius $r$ we have:
$$
\oint_C (2x-y^3)dx-xydy=
\o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1490401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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If $ \lim _{x\rightarrow -\infty }\sqrt {x^{2}+6x+3}+ax+b=1 $ then find $a+b$ $ \lim _{x\rightarrow -\infty }\sqrt {x^{2}+6x+3}+ax+b=1 $
if I use this $$\lim _{x\rightarrow -\infty }\sqrt {ax^{2}+bx+c}=\left| x+\dfrac {b} {2a}\right| $$
I find $a=1,b=4$ but if I try to multiple by its conjugate
$$\lim _{x\rightarrow -\... | Note that
$$
\lim _{x\rightarrow -\infty }\sqrt {ax^{2}+bx+c}=\left| x+\dfrac {b} {2a}\right|
$$
is meaningless, because the limit cannot depend on $x$.
One way can be doing a substitution, $x=-1/t$, so the limit becomes
$$
\lim_{t\to0^+}\sqrt{\frac{1}{t^2}-\frac{6}{t}+3}-\frac{a}{t}+b=
\lim_{t\to0^+}\frac{\sqrt{1-6t+3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1492778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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A limit problem about $a_{n+1}=a_n+\frac{n}{a_n}$ Let $a_{n+1}=a_n+\frac{n}{a_n}$ and $a_1>0$. Prove $\lim\limits_{n\to \infty} n(a_n-n)$ exists.
In my view, maybe we can use
$${a_{n + 1}} = {a_n} + \frac{n}{{{a_n}}} \Rightarrow {a_{n + 1}} - \left( {n + 1} \right) = \left( {{a_n} - n} \right)\left( {1 - \frac{1}{{{a_... | Let's start with some observations:
1) You can clearly see from what you have developed that $a_n\geq n$ (equality for $a_1=1$. For the rest we assume $a_1=1+\alpha>1$).
2) It is also easy to see that $a_{n}-n$ is decreasing. So if $a_1=\alpha+1$ then $a_n-n<\alpha$.
3) Consider the product $\prod\limits_{k = 1}^{n - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1493528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
If $1+2^n+4^n$ is prime number then prove that $n=3^k$ for some $k\in \mathbb{N}$. If $1+2^n+4^n$ is prime number then prove that $n=3^k$ for some $k\in \mathbb{N}$.
I've looked at https://math.stackexchange.com/a/186723/283318 for an inspiration. In base 2, $1+2^n+4^n=10\ldots010\ldots01$ and I am completely stuck.
| Let's solve a slightly more general problem and let's do it a novel way: Consider the function, for fixed $a$:
$$f_a(n)=1+a^n+a^{2n}.$$
We are here wondering about the case $a=2$. We will prove the following statement:
$f_a(1)$ divides $f_a(3n+1)$ and $f_a(3n+2)$ for any non-negative integer $n$.
This proves to be re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1494706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Recurrence Relation $T_n=\sum_{r=0}^{n-1} T_r+2^n$ From a recent solution I posted here, working from an alternative path would have led to the following recurrence relation which involves a summation term:
$$T_n=\sum_{r=0}^{n-1}T_r+2^n; \qquad T_0=1\qquad (n>1)$$
How can this be solved?
Edit 3 (replaces previous edits... | Use generating functions. Define $T(z) = \sum_{n \ge 0} T_n z^n$, adjust indices in the recurrence:
$\begin{align}
T_{n + 1} = \sum_{0 \le r \le n} T_r + 2 \cdot 2^r
\end{align}$
Multiply by $z^n$ and sum over $n \ge 0$, recognize some sums in the result:
$\begin{align}
\frac{T(z) - T_0}{z}
= \frac{T(z)}{1 -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Prove that $\cos(A) + \cos(B) = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$ I've seen this identity on examsolutions, but I'm unsure on how to prove it.
$$\cos(A) + \cos(B) = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$$
| Substitute the following: $v := \frac{x+y}{2}$ and $w := \frac{x-y}{2}$
Hence: $$\cos(x) + \cos(y) = \cos(v + w) + \cos(v - w)$$
Using the addition theorem of the cosinus yields:
$$= \cos(v) \cdot \cos(w) - \sin(v) \cdot \sin(w) + \cos(v) \cdot \cos(w) + \sin(v) \cdot \sin(w)$$
$$= 2 \cdot \cos(v) \cdot \cos(w) = 2 \cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1495966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
How to evaluate the definite integral by the limit definition $\int_{-1}^1 x^3 dx$? Solve the definite integral by the limit definition:
$$\int_{-1}^1 x^3 dx$$
The formula:
$$\int_a^bf(x)dx= \lim_{n\rightarrow \infty} \sum_{i=1}^n f(c_i)\Delta x_i$$
Get the variables:
$$\Delta x_i : \frac{b-a}{n} = \frac{1-(-1)}{n} = \... | You made an "algebra capital sin" when you wrote
$$
\left(-1+\frac{2i}n\right)^3=-1+\frac{8i^3}{n^3}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Diophantine equation: $y^2=1+12x+16x^2$ The diophantine equation $$y^2=1+12x+16x^2$$ only has solutions $x=0, y=\pm1$ according to wolfram alpha. How would I go about proving these are the only solutions?
Similarly the equation $$y^2=5+12x+16x^2$$ has solutions $x=-1, y=\pm3$.
Is there a general method with regards to ... | $1+12x+16x^2$ is closest to $(4x+1)^2$ and $(4x+2)^2$ regardless of $x\geq0$ or $x<0$, as it is easy to see the distance from $(4x)^2$ and $(4x+3)^2$ to $1+12x+16x^2$ are strictly larger. You need to check both equality and finds out it has integer solution only when $x=0$
Similarly,
$5+12x+16x^2$ is closest to $(4x+1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Show that $a = b = c = 0$ for $a\sqrt{2} + b = c\sqrt{3}$ is This is the following question:
Suppose that $a, b, c$ are integers such that
$a\sqrt{2} + b = c\sqrt{3}$
(i) By squaring both sides of the equation, show that $a = b = c = 0$
The answer says that you put the equation into the following form:
if $ab \neq 0$
$... | If $a,b \in \mathbb{Z}$, then
$$
2a^{2} + b^{2} + 2ab\sqrt{2} = 3c^{2},
$$
and then
\begin{align}
(*)\ \ \ \ ab\sqrt{2} = \frac{3c^{2}-2a^{2}-b^{2}}{2} .
\end{align}
The number $\sqrt{2}$ is irrational;
so $ab \neq 0$ leads to a contradiction,
and hence $ab = 0$.
We claim that $a=b=c = 0$; without loss of generality,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1498091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Find the complex number $z$ such that it satisfies: 1.$|z+\frac{1}{z}|=\frac{\sqrt{13}}{2} $, 2.$[Im (z)]^2+ [Re(z)]^2=2$ Find the complex number $z$ such that it satisfies: $$\left|z+\frac{1}{z}\right|=\frac{\sqrt{13}}{2} $$$$\Im (z)^2+ \Re(z)^2=2$$$$\frac{\pi}{2}<\arg(z)<\pi$$ then find $$z^{1991}$$
Know I was thinki... | Hint
Let $z=a+bi$ then
$$a^2+b^2=2,\dfrac{1}{z}=\dfrac{a-bi}{a^2+b^2}$$
and
$$|z+\dfrac{1}{z}|^2=\left(a+\dfrac{a}{a^2+b^2}\right)^2
+\left(b-\dfrac{b}{a^2+b^2}\right)^2=\dfrac{13}{4}$$
so
$$(a+\frac{a}{2})^2+(b-\frac{b}{2})^2=\frac{13}{4}$$
so
$$9a^2+b^2=13$$
then
$$a^2=\dfrac{11}{8},b^2=\dfrac{5}{8}$$
so
$$z=\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1498454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
If the coefficient of $x^{50}$ in the expansion of $(1+x)^{1000}+2x(1+x)^{999}+3x^2(1+x)^{998}$........ Problem :
If the coefficient of $x^{50}$ in the expansion of $(1+x)^{1000}+2x(1+x)^{999}+3x^2(1+x)^{998} +\cdots +1001x^{1000}$ is $\lambda$ then the value of $\frac{1952! 50!}{1001!}\lambda$
Please guide how to f... | Let $$\displaystyle S = (1+x)^{1000}+2x(1+x)^{999}+..1000x^{999}(1+x)+1001x^{1000}........(1)\times \frac{x}{(1+x)}$$
So we get $$\frac{x\cdot S}{(1+x)} = x(1+x)^{999}+2x^2(1+x)^{998}+.........+1000x^{1000}+\frac{1001x^{1001}}{1+x}....(2)$$
So we get $$S\left[1-\frac{x}{1+x}\right] = (1+x)^{1000}+\underbrace{\left[x(1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1500225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Infinitely many numbers that are the sum of two squares and the sum of two cubes, but...
I have to show that there are infinitely many natural numbers which
are simultaneously a sum of two squares and a sum of two cubes but
which are not a sum of two 6-th powers.
How to approach this question? Is there some simpl... | If we start with $n^3+5^3$ we have for sure a sum of two cubes. Since:
$$ n^3+5^3 = (n+5)((n-3)^2+(n+16)) $$
if we set $n=m^2-16$ we have:
$$ (m^2-16)^3 + 5^3 = (m^2-11)((m^2-19)^2+m^2) $$
and if we set $m=l+6$ we have:
$$ (l^2+12 l +20)^3 + 5^3 = ((l+5)^2+2l)((l^2+12 l+17)^2+(l+6)^2) $$
and if $l$ is twice a square, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1500463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Let $p> 7$, prove that $\left(\frac{2}{q}\right) = (-1)^{\frac{q^2-1}{8}}$ with $q$ an odd prime Let $p> 7$, prove that $\left(\frac{2}{q}\right) = (-1)^{\frac{q^2-1}{8}}$. with $q$ an odd prime. We can by using the following verifications:
$$\left(\frac{2}{p}\right) = \left(\frac{8-p}{p}\right) = \left(\frac{p}{p-8}\... | Hint : For a prime $p>2$, we have $$(\frac{2}{p})=1$$, if and only if $$p\equiv \pm1\ (\ mod\ 8)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Dominated Convergence Is there a function that dominates
$$f_n(x) = \frac{1}{(1+\frac{x}{n})^nx^{\frac{1}{n}}}$$
on $(1,\infty)$ for all $n$? I need to apply DCT to get $e^{-x}$.Obviously MCT doesn't apply since $x^{\frac{1}{n}}$ is decreasing but $(1+\frac{x}{n})^n$ is increasing to $e^x$. The best I got was $\frac{1... | First, we have for $x\in[1,\infty)$
$$x^{-1/n}\le 1 \tag 1$$
Next, we show that $\left(1+\frac xn\right)^n$ is increasing function of $n$ for $x>0$. To that end, we analyze the ratio
$$\begin{align}
\frac{\left(1+\frac x{n+1}\right)^{n+1}}{\left(1+\frac xn\right)^n}&=\left(\frac{n(n+1+x)}{(n+1)(n+x)}\right)^{n+1}\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1502841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
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