Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Matrix transformation shearing the triangle to be right triangle If we have a triangle at $(1,1), (5,3), (7,1)$, how to find the sheared matrix to transform the triangle to be right triangle at $(1,1)$.
Is it that we need to find $i$ in
$\begin{pmatrix}
1 && i \\
0 && 1 \\
\end{pmatrix}$
But how to ensure the right tri... | Definition
Coordinate Transformation
Given a 2D Point $(x,y)$, to transform the point to another 2D coordinate space is defined through the following set of equations
$$x'=a\cdot x + b\cdot y$$
$$y'=c\cdot c + d\cdot y$$
where $a,b,c,d$ are real value constant with the constraint $a\cdot d - b\cdot c \ne 0$
And can be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Help me to prove this integration Where the method used should be using complex analysis.
$$\int_{c}\frac{d\theta}{(p+\cos\theta)^2}=\frac{2\pi p}{(p^2-1)\sqrt{p^2-1}};c:\left|z\right|=1$$
thanks in advance
| i do my self like this
on $|z| = 1, z = e^{i\theta}, d\theta=\frac{dz}{iz}$
using substitution $\cos\theta=\frac{z+z^{-1}}{2}$
$\frac{1}{i}\int\frac{\frac{dz}{z}}{(p+\frac{z+{z}^{-1}}{2})(p+\frac{z+{z}^{-1}}{2})} or \frac{1}{i}\int\frac{1}{(2pz+z^2+1)(p+\frac{z+{z}^{-1}}{2})} $
and then multiple by $\frac{z}{z}$
$\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$.
Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
Here's my idea:
$\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
$2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(ab + bc + ca)$
$2(\sqrt{a} + \sqrt... | From the given inequality $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ observe that $$2(ab+bc+ac)=(a+b+c)^2-a^2-b^2-c^2$$ We can rewrite the original inequality as
$$a^2+2\sqrt{a}+ b^2+2\sqrt{b}+ c^2+2\sqrt{c}\ge9$$ since $(a+b+c)=3$. Using AM-GM
set the LHS up as follows:
$$a^2+\sqrt{a}+\sqrt{a}\ge3\sqrt[3]{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/336362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 4,
"answer_id": 0
} |
A problem with limit How to attack this one?
Does the following limit exists:
$$\lim_{x\to +\infty}\dfrac {\cos^5x\sin^5x} {x^8\sin^2x-2x^7\sin x\cos^2x+x^6\cos^4x+x^2\cos^8x}$$
| The denominator can be rewritten as the sum of the two nonnegative terms
$$
(x \sin x - \cos^2 x)^2 x^6 + x^2 \cos^8 x.
$$
For all integral $n\ge 1$, when $x$ is in the interval $[2\pi n-\frac{\pi}{4}, 2\pi n+\frac{\pi}{4}]$ or $[2\pi n+\frac{3\pi}{4}, 2\pi n+\frac{5\pi}{4}]$, $|\cos x|\ge 2^{-1/2}$, so the second term... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337081",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Convergence of the sequence $x_{k+1}=\frac{1}{2}(x_k+\frac{a}{x_k})$
Consider the sequence $x_{k+1}=\frac{1}{2}(x_k+\frac{a}{x_k}), a\gt 0, x\in\mathbb{R}$. Assume the sequence converges, what does it converge to?
I'm having trouble seeing how to start,
Any help would be appreciated
Thanks
| This sequence has a closed form.
First, if $x_0=\sqrt{a}$, $x_n=\sqrt{a}$ for all $n$, and the sequence is constant. We will thus assume $x_0\ne\sqrt{a}$ in the following.
Now, given $x_n\ne\sqrt{a}$ for some $n$, we have
$$x_{n+1}-\sqrt{a}=\frac{1}{2}\left(x_n+ \frac{a}{x_n} \right) - \sqrt{a}=\frac{1}{2}\left( \sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Is $\lim_{n\to +\infty} \frac{\sqrt{n+1}}{2n^2+1}$ convergent? I don't know how to simplify this expression to apply the ratio test to determine if the sequence converges or diverges
$$\lim_{n\to +\infty} \frac{\sqrt{n+1}}{2n^2+1}$$
| Estimate sequence terms above and below
$$\frac{1}{2(n+1)^{\frac{3}{2}}}=\frac{\sqrt{n+1}}{2(n+1)^2} \leqslant \frac{\sqrt{n+1}}{2n^2+1} \leqslant \frac{2\sqrt{n}}{2n^2}=\frac{\sqrt{n}}{n^2}=\frac{1}{n^{\frac{3}{2}}}.$$
Hence $$\frac{\sqrt{n+1}}{2(n+1)^2}=O({n^{-\frac{3}{2}}}), \;\;\;n\to\infty.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/338336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Find a closed form of the series $\sum_{n=0}^{\infty} n^2x^n$ The question I've been given is this:
Using both sides of this equation:
$$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$
Find an expression for $$\sum_{n=0}^{\infty} n^2x^n$$
Then use that to find an expression for
$$\sum_{n=0}^{\infty}\frac{n^2}{2^n}$$
This is a... | We can write any monomial, or polynomial in $n$, as a falling factorial polynomial using Stirling numbers of the second kind or this simple algorithm.
After we can use the following identities of finite calculus
$$\Delta_n n^\underline m=(n+1)^\underline m- n^\underline m=m n^\underline{m-1},\quad \sum n^\underline m\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 2
} |
Solving linear recurrence relation Solve the following linear recurrence relation:
$$h_n=4h_{n-1}-4h_{n-2}+n^2 2^n$$
for $n\geq2$ and $h_0=h_1=1$
| $$h_n - 2h_{n-1} = 2(h_{n-1} - 2h_{n-2}) + n^2 \cdot 2^n$$
Let $h_n - 2h_{n-1} = f_n$. We then have
$$f_n = 2f_{n-1} + n^2 \cdot 2^n = 2(2f_{n-2} + (n-1)^2 \cdot 2^{n-1}) + n^2 \cdot 2^n = 4 f_{n-2} + 2^n \cdot (n^2 + (n-1)^2)$$
Hence, by induction, we have
$$f_n = 2^n \cdot (n^2 + (n-1)^2 + \cdots + 2^2 + 1^2) + 2^n \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
$\lim _{n \rightarrow \infty }a_{n}=\sqrt{a_{n-2}a_{n-1}}$ , $a_1=1, a_2=2$ I was asked to find the limit of:
$a_{n+2}=\sqrt{a_na_{n+1}}$
$a_1=1, a_2=2$
It seems as if the sequence is constant from n=4 and it's value is $a_n=\sqrt{2\sqrt{2}} -\forall{n>3}$
I'd just like to double-check I did thing right.
Th... | Take logs of both sides and define $b_n = \log{a_n}$. Then
$$2 b_n - b_{n-1}-b_{n-2} = 0$$
$$b_0=0$$
$$b_1=\log{2}$$
This is a constant coefficient difference equation with solution $b_n=A r^n$,where $r$ satisfies
$$2 r^2-r-1=0$$
with solutions $r_+ = 1$ and $r_-=-1/2$. Thus
$$b_n = A + B \left (-\frac{1}{2} \right )... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/340408",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$ While extending my calculation techniques, with aid of Mathematica, I found that
\begin{align*}
\int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx
&= -6 \zeta '(-1) -\frac{19}{24}, \\
\int_{0}^{... | You can try
$$
\int_0^1\left(\frac{1}{\log x}+\frac{1}{1-x}\right)^m (-\log x)^{s-1}\,dx
$$
for $s$ with sufficiently large real part.
This will give you an expression involving $\Gamma$ and $\zeta$ functions.
Then use the analytic continuation to $\sigma>0$ and plug in $s=1$.
I checked this method for $m=2$ and got... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/340718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 1,
"answer_id": 0
} |
Does a function exist with the property $f(-n^2+3n+1)=(f(n))^2+1$? Let $f:\mathbb{R}\to \mathbb{R}$ be a function which fulfills for every $n \in \mathbb{N}$
$$f(-n^2+3n+1)=(f(n))^2+1$$
Is it possible that such a function exists?
| The main idea here is writing at first some of the equations you get and look if they have common terms. Indeed here the terms for $n=3$ and $n=1$ are very interessting, as in both only occur $f(1)$ and $f(3)$.
\begin{align*}
f(3)&=1+f(1)^2 \tag{$i$}\\
f(1)&=1+f(3)^2 \tag{$ii$}\\
\end{align*}
As we don't know that muc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/341479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Factor $(a^2+2a)^2-2(a^2+2a)-3$ completely I have this question that asks to factor this expression completely:
$$(a^2+2a)^2-2(a^2+2a)-3$$
My working out:
$$a^4+4a^3+4a^2-2a^2-4a-3$$
$$=a^4+4a^3+2a^2-4a-3$$
$$=a^2(a^2+4a-2)-4a-3$$
I am stuck here. I don't how to proceed correctly.
| If you assign $$ a^2 + 2a = x $$ you'll get: $$ x^2 - 2x - 3 $$
Considering that $$ x^2 - 2x - 3 = (x - 3)(x+1) $$ you'll get: $$ (a^2 + 2a - 3)(a^2 + 2a+1) = (a + 3)(a - 1)(a + 1)^2 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/342581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 0
} |
Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$ $$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$
I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then simplifying till I replicated the identity on the right. I am not s... | Here is an interesting different way: Let
$$x = \cos(A+B)\cos(A-B) \\
y = \sin(A+B)\sin(A-B)$$
Then,
$$x+y = \cos(A+B-A+B) = \cos(2B) \\
x-y = \cos(A+B+A-B) = \cos(2A)$$
You can add these to get $2x$ or subtract them to get $2y$. Then expand using the double-angle formulas. This gives you two trig product formulas at t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/345703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 3
} |
Conditional probabilities, urns I found this question interesting and apparently it has to do with conditional probabilities:
An urn contains six black balls and some white ones. Two balls are drawn simutaneously. They have the same color with probability 0.5. How many with balls are in
the urn?
As far as I am concerne... | probability of 2 black balls: $\frac{6}{8}*\frac{5}{7}=\frac{30}{56}$
of 2 white balls: $\frac{2}{8}*\frac{1}{7}=\frac{2}{56}$
$\frac{30}{56}+\frac{2}{56}=\frac{32}{56}\neq\frac{1}{2}$
Assume the number of white balls is $n$:
2 black balls = $\frac{6}{6+n}*\frac{5}{5+n}=\frac{30}{n^2+11n+30}$
2 white balls = $\frac{n}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/347187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Integral solutions of hyperboloid $x^2+y^2-z^2=1$ Are there integral solutions to the equation $x^2+y^2-z^2=1$?
| the equation:
$X^2+Y^2=Z^2-1$
Solutions can be written using the solutions of Pell's equation: $p^2-2k(k-1)s^2=1$
$k$ - given by us.
Solutions hav e form:
$X=2kps-2(k-1)s^2$
$Y=2(k-1)ps+2ks^2$
$Z=p^2+2(k^2-k+1)s^2$
And more:
$X=2p^2-2(3k-2)ps+2(2k-1)(k-1)s^2$
$Y=2p^2-2(3k-1)ps+2k(2k-1)s^2$
$Z=3p^2-4(2k-1)ps+2(3k^2-3k+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/351491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 9,
"answer_id": 2
} |
To find Area of rectangular with given 3 parameters
$a,b,c$ are given parameters . I would like to find Area of (ABCD) rectangular.
I can find $d$ from $a,b,c$.
$$(x-m)^2+(y-n)^2=a^2$$
$$(x-m)^2+n^2=b^2$$
$$m^2+(y-n)^2=c^2$$
$$m^2+n^2=d^2$$
$$m^2+n^2+(x-m)^2+(y-n)^2=a^2+d^2=b^2+c^2$$
$$d=\sqrt {b^2+c^2-a^2}$$
Let's d... |
look at the above picture, you can see orange and yellow rectangles are all satisfy a,b,c if m,n is not fixed. and there are many such rectangles so you can't find the area. but you can find max area by a,b,c which might be another exercise you can do which is very hard to find out the final result.
for m,n is fixed, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/351854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Calculating the integral $\int_{0}^{\pi /6}\sqrt{1-\left(\frac{R_s\sin \theta }{C_L}\right)^2} d\theta$ I want to integrate $I=\int\limits_{0}^{\pi /6}{\sqrt{1-{{\left( \frac{{{R}_{s}}\sin \theta }{{{C}_{L}}} \right)}^{2}}}d \theta}$.
I get incomplete elliptic integral $E(z\mid m)$ in the calculation by mathematica. I ... | For the binomial series of $\sqrt{1-x}$ , $\sqrt{1-x}=\sum\limits_{n=0}^\infty\dfrac{(2n)!x^n}{4^n(n!)^2(1-2n)}$
$\therefore\int_0^{\frac{\pi}{6}}\sqrt{1-\left(\dfrac{R_s\sin\theta}{C_L}\right)^2}~d\theta=\int_0^{\frac{\pi}{6}}\sum\limits_{n=0}^\infty\dfrac{(2n)!R_s^{2n}\sin^{2n}\theta}{4^n(n!)^2(1-2n)C_L^{2n}}d\theta=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/352043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How to integrate $f( \theta ) = \frac{1}{a + \sin( \theta ) }$? Let $a > 1$. I am wondering how evaluate the integral: $$ \int_{0}^{2 \pi } \frac{1}{a + \sin( \theta) } d \theta $$ by means of methods of complex analysis. In the homework assignment, the following hint is given: write $\sin( \theta ) = (e^{i \theta } - ... | As you did, let $z=e^{i\theta}$. Then $dz=izd\theta$ and
$$ \sin\theta=\frac{1}{2i}\left(z-\frac{1}{z}\right) $$
and hence
\begin{eqnarray*}
\int_0^{2\pi}\frac{1}{a+\sin\theta}d\theta&=&\int_{|z|=1}\frac{1}{a+\frac{1}{2i}\left(z-\frac{1}{z}\right)}\frac{dz}{iz}\\
&=&2\int_{|z|=1}\frac{1}{z^2+2aiz-1}dz\\
&=&2\int_{|z|=1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Finding the maximum of $\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}$ If $a,b,c,d$ are distinct real numbers such that $\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}=4$ and $ac=bd$.
Then how would we calculate the maximum value of $$\dfrac{a}{c}+\dfrac{b}{d}+\dfrac{c}{a}+\dfrac{d}{b}.$$
I was unable to proceed... | Let $w=\frac{a}{b}$, $x=\frac{b}{c}$, $y=\frac{c}{d}$, $z=\frac{d}{a}$. Then
$$\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}=wx+xy+yz+zw=(x+z)(w+y)\leqslant\Bigl(\frac{w+x+y+z}{2}\Bigr)^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/358223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate? I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident.
In particular, Ramanujan's does the following compari... | After this step:
$$
\lfloor\frac{x}{b_1}\rfloor + 1 - \frac{x}{b_1} \ge (\lfloor\frac{x}{b_2}\rfloor + 1 - \frac{x}{b_2} - \frac{1}{2}) + (\lfloor\frac{x}{b_3}\rfloor + 1 - \frac{x}{b_3} - \frac{1}{2})
$$
we have
$$
\lfloor\frac{x}{b_1}\rfloor + 1 \ge (\lfloor\frac{x}{b_2}\rfloor + 1) + (\lfloor\frac{x}{b_3}\rfloor + 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Basis of complex matrix vector space over $\Bbb{R}$ I understand that the basis of the vector space $$Mat_2(\Bbb{R}) = \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$$ over $\Bbb{R}$ is $$e = \left\{ \begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix},\begin{pmatrix}0 & ... | That would be $e = \left\{ \begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix},\begin{pmatrix}0 & 0\\ 1 & 0\end{pmatrix},\begin{pmatrix}0 & 0\\ 0 & 1\end{pmatrix} \right\}$together with $e' = \left\{ \begin{pmatrix}i & 0\\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & i\\ 0 & 0\end{pmatrix},\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/360599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\tan(75^\circ) = 2 + \sqrt{3}$ My (very simple) question to a friend was how do I prove the following using basic trig principles:
$\tan75^\circ = 2 + \sqrt{3}$
He gave this proof (via a text message!)
$1. \tan75^\circ$
$2. = \tan(60^\circ + (30/2)^\circ)$
$3. = (\tan60^\circ + \tan(30/2)^\circ) / (1 - \ta... | A proof without words (but it uses some geometry). Is that OK?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/360747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 0
} |
Linear algebra on Fibonacci number Consider the sequence $\{a_n\}_{n\ge 0}$ given by the recurrence relation
$$a_0=1,\ a_1=-1,\ a_{n+1}=3a_n+10a_{n-1}\ \ \text{for } n\ge2$$
And I am asked to work out the closed form expression for an in the same fashion as the proof for Fibonacci numbers by using linear algebra way
Th... | We can use the generating function $f(x)$ of $\{a_n\}$ to solve. Let $f(x)=\sum_{n=0}^\infty a_nx^n$. Then
\begin{eqnarray*}
f(x)&=&\sum_{n=0}^\infty a_nx^n=1-x+\sum_{n=2}^\infty a_nx^n\\
&=&1-x+\sum_{n=1}^\infty a_{n+1}x^{n+1}\\
&=&1-x+\sum_{n=1}^\infty (3a_{n}+10a_{n-1})x^{n+1}\\
&=&1-x+3\sum_{n=1}^\infty a_{n}x^{n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
$c$ is a complex number that satisyfing $(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$ Let $c$ is complex-number satisfying :
$(c+\frac{1}{c}+1)(c+\frac{1}{c}) = 1$
So, how could i get
$(3c^{100}+\frac{2}{c^{100}}+1)(c^{100}+\frac{2}{c^{100}}+3)$ ?
| HINT:
On simplification, $c^4+c^3+c^2+c+1=0$
Clearly, $c\ne1$
Multiply either sides the $(c-1),$ we get $c^5-1=(c-1)\cdot0=0$
$\implies c^5=1\implies c^{100}=(c^5)^{20}=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/364214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Summing $ \sum _{k=1}^{n} k\cos(k\theta) $ and $ \sum _{k=1}^{n} k\sin(k\theta) $ I'm trying to find
$$\sum _{k=1}^{n} k\cos(k\theta)\qquad\text{and}\qquad\sum _{k=1}^{n} k\sin(k\theta)$$
I tried working with complex numbers, defining $z=\cos(\theta)+ i \sin(\theta)$ and using De Movire's, but so far nothing has come ... | All identities are coming from Wikipedia page for trigonometric identities. The main identity that we start with is:
$$
\sum_{k=1}^n \cos(kx)= \frac{ \sin((n+\frac{1}{2}) x)}{2\sin(x/2)} -1
$$
Get a derivative with respect to $x$, and we have
\begin{eqnarray*}
\sum_{k=1}^n k \sin(kx)&=& \frac{ \cos(x/2) \sin((n+\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/364631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
A counting problem involving ternary sequences
A ternary sequence is a sequence all of whose elements are the digits 0, 1 or 2. Find the number of ternary sequences of length 8 in which the digits 0 and 1 each occur an even number of times.
The first case that should be considered is when we have all 2s, in which ... | Do as follows: Define $a_n$ as the number of $n$-sequences with an even number of 0 and an even number of 1, $b_n$ for even/odd, $c_n$ for odd/even and $d_n$ for odd/odd. Then $a_0 = 1$, and $b_0 = c_0 = d_0 = 0$. You can set up recurrences for all for them (think how you can get e.g. odd/even by adding a symbol):
$$
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Symmetry properties of $\sin$ and $\cos$. Why does $\cos\left(\frac{3\pi}{2} - x\right) = \cos\left(-\frac{\pi}{2} - x\right)$? For a question such as:
If $\sin(x) = 0.34$, find the value of $\cos\left(\frac{3\pi}{2} - x\right)$.
The solution says that:
\begin{align*}
\cos\left(\frac{3\pi}{2} - x\right) &= \cos\left... | Remember that cosine is $2\pi$ periodic this. This means that
$$\cos(x + 2n\pi) = \cos(x)$$
for any $x\in\mathbb{R}$ and for any integer $n\in\mathbb{Z}$. Basically, this says that if you can shift the graph of cosine left or right by $2\pi$ without changing it. Shifting it once gives you your desired equality since
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Is there a nice way to interpret this matrix equation that comes up in the context of least squares So I am working on this problem with fitting a second degree polynomial of the form $y=a_1x^2+a_2x+a_3$ to four points using least squares. One of the parts of the problem is to write out the matrix equation that describ... | To augment the insightful post of @Christopher A. Wong, we offer two other perspectives.
First, why do we form the normal equations? To form a consistent linear system. Typically a linear algebra course begins with problems like
$$
\mathbf{A} a - y = 0.
$$
We learn Gaussian elimination, $\mathbf{L}\mathbf{U}$ decomposi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to integrate this integral
Let $$f\left(x\right)= \mbox{ the antiderivative } \frac{x^2}{1-x^5}, $$
$f\left(1\right)=0$
Find $f\left(4\right)$
I know that:
$F\left(x\right) = I\left(x\right) + C $ where $I\left(x\right)$ is the antiderivative (integral) of $f\left(x\right)$
Thus at $x = 1\; F\left(1\right) = 0 = ... | Equation $z^5-1=0$ has one real root $z_0=1$ and two pairs of complex-conjugated roots $z_k=e^{\tfrac{2k\pi i}{5}},\;\; k=1,\,\ldots\,,4.$ Then
$$
z^5-1=(z-z_0)(z-z_1)(z-\bar{z}_1)(z-z_2)(z-\bar{z}_2)=\\
=(z-1)(z^2-2\Re{(z_1)} z+1)(z^2-2\Re{(z_2)} z+1)=\\
=(z-1)\left(z^2-2\cos{\left(\dfrac{2\pi}{5}\right)} z+1\right)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/370122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
What is going on with this constrained optimization? I'd like to figure out what is going on when trying to maximize a function (below $a_i$ are real numbers)
$F = a_1a_2 + a_2a_3 + \cdots + a_{n - 1}a_n + a_na_1;$
When we have active constraints
$h_1 = a_1 + a_2 + \cdots + a_n = 0;$
$h_2 = a_1^2 + a_2^2 + \cdots +a_n... | Consider what happens for $n=2$ and $n=3$.
$n=2$: Only two points satisfy the constraints, $(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})$ and $(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$, and $F$ takes the same value on these two points.
$n=3$: The constraints describe a circle of radius 1 contained in the plane perpendicu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/370251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is the polar form of $ z = 1- \sin (\alpha) + i \cos (\alpha) $? How do I change $ z = 1- \sin (\alpha) + i \cos (\alpha) $ to polar? I got $r = (2(1-\sin(\alpha))^{\frac{1}{2}} $. I have problems with the exponential part. What should I do now?
| $1-\sin\alpha\ge 0$
If $1-\sin\alpha=0,\cos\alpha=0, z=0+i\cdot0$
If $1-\sin\alpha> 0$
$1-\sin\alpha=(\cos\frac\alpha2-\sin \frac\alpha2)^2$ and
$\cos\alpha=\cos^2\frac\alpha2-\sin^2\frac\alpha2$
So, $z=(1-\sin\alpha)+i(\cos\alpha)=(\cos\frac\alpha2-\sin \frac\alpha2)^2+i(\cos^2\frac\alpha2-\sin^2\frac\alpha2)$
If $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/371960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Prove That $x=y=z$ If $x, y,z \in \mathbb{R}$,
and if
$$ \left ( \frac{x}{y} \right )^2+\left ( \frac{y}{z} \right )^2+\left ( \frac{z}{x} \right )^2=\left ( \frac{x}{y} \right )+\left ( \frac{y}{z} \right )+\left ( \frac{z}{x} \right ) $$
Prove that $$x=y=z$$
| A similar triple completing-the-square is helpful. Using your notation, $(a-1)^2+(b-1)^2+(c-1)^2+2(a+b+c)-3=a+b+c$. This rearranges to $(a-1)^2+(b-1)^2+(c-1)^2=3-(a+b+c)$. Hence $a+b+c\le 3$ since otherwise the sum of three squares would be negative. Consequently $a^2+b^2+c^2\le 3$.
Unfortunately, I don't have a cu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/372143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 3
} |
Computing the unit normal vector - Simplifying help I have a surface
$$X(u,v) = \left(3uv^2 - u^3 - \frac{u}{3}, 3u^2v - v^3 - \frac{v}{3}, 2uv \right), $$
and the cross product
$$(X_u \times X_v) = \left(3(u^2 + v^2) \frac{1}{3} \right) \cdot \left(2v, 2u, \frac{1}{3} - 3(u^2 + v^2) \right).$$
I have to show that the ... | Expanding $(3(u^{2} + v^{2}) + \frac{1}{3})^{2}$, we see:
$(3(u^{2} + v^{2}) + \frac{1}{3})*(3(u^{2} + v^{2}) + \frac{1}{3}) = 9(u^{2}+v^{2})^{2} + 2*3*\frac{1}{3}(u^{2} + v^{2}) + \frac{1}{9}$;
Does this help?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/372741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving quadratic equations by completing the square.
Graphing $y=ax^2+ bx + c$ by completing the square
*
*Add and subtract the square of half the coefficent of $x$.
*Group the perfect square trinomial.
*Write the trinomial as a square of a binomial.
Rewrite $y = x^2 + 6x + 8$ into $y = a(x-h)^2 + k$.
I've tri... | To complete the square for the equation:
$y = x² + 6x + 8$
Start by:
1) $x^2+6x+8$
2) $x^2+\dfrac 62x +8$ *divide the bx by 2
3) $(x^2+\dfrac 62x + (\dfrac 62)^2)+8$ *square the bx term to form a perfect square
4) $(x^2+ 3x + (36/4))+8$
5) $(x^2+3x + 9)+8-9$ *add the inverse of the c term
6) $(x^2+3x+9)-1$ *factor the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/375833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
If $S_n = 1+ 2 +3 + \cdots + n$, then prove that the last digit of $S_n$ is not 2,4 7,9. If $S_n = 1 + 2 + 3 + \cdots + n,$ then prove that the last digit of $S_n$ cannot be 2, 4, 7, or 9 for any whole number n.
What I have done:
*I have determined that it is supposed to be done with mathematical induction.
*The formul... | Note that $1+2+3+4+5$ is divisible by $5$, and for the same reason $5k+1+5k+2+5k+3+5k+4+5k+5$ is divisible by $5$ for any integer $k$.
Let $5m$ be the largest multiple of $5$ which is $\le n$. Then $n=5m$ or $n=5m+1$ or $n=5m+2$ or $n=5m+3$ or $n=5m+4$.
The sum of the integers from $1$ to $5m$ is divisible by $5$. So w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Expectation and children What is the expected number of children if the probability of having a child(boy or girl) = 1/2 and if you want a boy and a girl?
E(No of children) = ?
| Whatever sex the first-born kid is, our waiting time after that until a child of the opposite sex is born is $\frac{1}{1/2}=2$. So the expected number of children is $3$.
For more detail, we can assume without loss of generality that the first-born is a boy. Let $y$ be the expected additional number of children until ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is $1/z$ differentiable on $\mathbb{C}\setminus\{0\}$? The way I usually solve these kind of questions is by using the Cauchy Riemann Equations...
$h(z)=\dfrac{1}{z}$
$=\dfrac{1}{x+iy}$
$=\dfrac{1}{x+iy} \dfrac{x-iy}{x-iy}$
$=\dfrac{x-iy}{(x+iy)(x-iy)}$
$=\dfrac{x-iy}{x^2+y^2}$
$=\dfrac{x}{x^2+y^2}-\dfrac{iy}{x^2+y^2}$... | Hint:
$$\frac{1}{x+iy}\not=\frac{1}{x}+\frac{1}{iy}.$$
However,
$$
\frac{1}{x+iy}=\frac{1}{x+iy}\frac{x-iy}{x-iy}
$$
Does this help?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/379280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Approximating a log-power function I can't figure out how the following approximation has been done, I would appreciate any guidance:
$$y=-60+10\log_{10}\left[\frac{\left(\frac{99}{100}\right)^m}{\frac{1}{11}\left(\frac{1}{3}\right)^m+\frac{10}{11}\left(\frac{1}{6}\right)^m}\right]$$
is approximated to: $y=-49.6+4.73m$... | Let's put $y$ in logarithm and exponential form (factorizing $\left(\frac 16\right)^m=\left(\frac 13\right)^m\left(\frac 12\right)^m$) :
\begin{align}
y&=-60+\frac {10}{\ln(10)}\;\ln\left[\frac{e^{m\ln\left(\frac{99}{100}\right)}}{\frac{1}{11}e^{m\ln\left(\frac{1}{3}\right)}\left(1+\frac {10}{2^m}\right)}\right]\\
&=-6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/381934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find no. of points where $f$ and $g$ meet.
If $f(x)=x^2$ and $g(x)=x \sin x+ \cos x$ then
(A) $f$ and $g$ agree at no points
(B) $f$ and $g$ agree at exactly one point
(C) $f$ and $g$ agree at exactly two points
(A) $f$ and $g$ agree at more than two points.
Trial: I think I need to find the values of $x$ for which $... | Note that $x^2$ and $x\sin x+\cos x$ are both even functions, so the graphs of $y=x^2$ and $y=x\sin x+\cos x$ are symmetric about the $y$ axis. So if we kow the story for $x\ge 0$, we know the story everywhere.
At $x=0$, the curve $y=x^2$ is below the curve $y=x\sin x+\cos x$.
But $x^2$, in the long run, is far bigger ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What have I done wrong in solving this problem with indices rules? The question asks to simplify:
$$\left(\dfrac{25x^4}{4}\right)^{-\frac{1}{2}}.$$
So I used $(a^m)^n=a^{mn}$ to get
$$\dfrac{25}{4}x^{-2} = \dfrac{25}{4} \times \dfrac{1}{x^2} = \frac{25}{4x^2} = \frac{25}{4}x^{-2}$$
However, this isn't the answer, and ... | $$\Big(\frac{25x^4}{4}\Big)^{-\frac{1}{2}}=\Big(\frac{4}{25x^4}\Big)^{\frac{1}{2}}=\frac{4^{\frac{1}{2}}}{25^{\frac{1}{2}}(x^4)^{\frac{1}{2}}}=\frac{2}{5x^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/388748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Reducibility of $x^{2n} + x^{2n-2} + \cdots + x^{2} + 1$ Just for fun I am experimenting with irreducibility of certain polynomials over the integers. Since $x^4+x^2+1=(x^2-x+1)(x^2+x+1)$, I thought perhaps $x^6+x^4+x^2+1$ is also reducible. Indeed:
$$x^6+x^4+x^2+1=(x^2+1)(x^4+1)$$
Let $f_n(x)=x^{2n}+x^{2n-2}+\cdots + ... | I am a bit surprised that the answers given so far that invoke cyclotomic polynomials don't go all the way to exhibit a factorisation of $1+x^2+\cdots+x^{2n}$. Using $x^m-1=\prod_{d\mid m}\Phi_d(x)$ one gets
$$
1+x^2+\cdots+x^{2n}=\frac{x^{2n+2}-1}{x^2-1}=\prod_{\substack{d\mid 2(n+1) \\ d\notin\{1,2\}}}\Phi_d(x),
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/391086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 3
} |
Solve $\sqrt{2x-5} - \sqrt{x-1} = 1$ Although this is a simple question I for the life of me can not figure out why one would get a 2 in front of the second square root when expanding. Can someone please explain that to me?
Example: solve $\sqrt{(2x-5)} - \sqrt{(x-1)} = 1$
Isolate one of the square roots: $\sqrt{(2x-5... | You can solve the square of a sum by writing out the square as a product of two sums and writing out the multiplication for each pair of terms.
$$(1+\sqrt{x-1})^2 =\\
(1+\sqrt{x-1})(1+\sqrt{x-1})=\\
1\times1 + 1\times\sqrt{x-1}+\sqrt{x-1}\times1+\sqrt{x-1}\times\sqrt{x-1}=\\
1+\sqrt{x-1}+\sqrt{x-1}+x-1=\\
2\sqrt{x-1}+x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
} |
Find the value of $\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$ I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
| My calculation shows that
\begin{align*}
\int_{0}^{\infty} \frac{x^3}{(x^4 + 1)(e^x - 1)} \, dx
&= \frac{\gamma}{2} - \log\sqrt{2\pi} + \frac{\pi}{4} \frac{\sin\frac{1}{\sqrt{2}}}{\cosh\frac{1}{\sqrt{2}} - \cos\frac{1}{\sqrt{2}}} \\
&\quad - \frac{1}{2}\sum_{n=1}^{\infty} \frac{1}{n\left(1 + (2\pi n)^4\right)} \\
&\app... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "40",
"answer_count": 3,
"answer_id": 1
} |
How to solve this integral for a hyperbolic bowl? $$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
| A related problem. Note that,
$$ z=\sqrt{ 1+x^2+y^2 } \implies z_x=\frac{x}{\sqrt{ 1+x^2+y^2 }},\quad z_y=\frac{y}{\sqrt{ 1+x^2+y^2 }} $$
$$ \iint_{s} z dS = \iint_{D} z \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2} dA $$
$$ = \iint_{D} \sqrt{1+(x^2+y^2)}\sqrt{{\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/393386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$ For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series
$$
\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n.
$$
For $b=0$, this post shows
$$
\sum_{n=0}^\infty \binom{2n}{n}x^n = (1-4x)^{-1/2}.
$$... | Note that the quoted identity is not difficult to verify using a
variant of Lagrange Inversion. Introduce
$$T(z) = w = \sqrt{1-4z}$$ so that
$$z = \frac{1}{4} (1-w^2)$$ and
$$dz = -\frac{1}{2} w \; dw$$
Then we seek to compute
$$[z^n] \frac{2^b}{T(z)} (1+T(z))^{-b}
= \frac{1}{2\pi i}
\int_{|z|=\epsilon} \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/393616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Expressing $\sqrt{n +m\sqrt{k}}$ Following this answer, is there a simple rule for determining when:
$$\sqrt{n +m\sqrt{k}}$$
Where $n,m,k \in \mathbb{N}$, can be expressed as:
$$a + b\sqrt{k}$$
For some natural $a,b$?
This boils down to asking for what $n,m,k \in \mathbb{N}$ there exist $a,b\in \mathbb{N}$ such that:
... | Hint: Please see this link. To translate it to your problem, note that $ b = m^2k $. From here, one of the square roots on the right hand side of their expression must reduce.
In the case that $ a + \sqrt{a^2 - b} = 2p^2, p \in \mathbb{N} $, $ p^2 = \frac{a + \sqrt{a^2 - b}}{2} $. For the second case $ q^2 = \frac{a -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/394056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to prove this inequality $xy\sin^2C+yz\sin^2A+zx\sin^2B\le\dfrac{1}{4}$ Let $x,y,z$ is real numbers,and such that $x+y+z=1$,and in $\Delta ABC$,prove that
$$xy\sin^2C+yz\sin^2A+zx\sin^2B\le\dfrac{1}{4}$$
I think this inequality maybe use $x^2+y^2+z^2\ge 2yz\cos{A}+2xz\cos{B}+2xy\cos{C},x,y,z\in R$
Thank you everyon... | We need to prove that $$(x+y+z)^2\geq4xy\sin^2\gamma+4xz\sin^2\beta+4yz\sin^2\alpha$$ or
$$x^2+y^2+z^2+2xy\cos2\gamma+2xz\cos2\beta+2yz\cos2\alpha\geq0$$ or
$$z^2+2(x\cos2\beta+y\cos2\alpha)z+x^2+y^2+2xy\cos2\gamma\geq0,$$
for which it's enough to prove that
$$(x\cos2\beta+y\cos2\alpha)^2-(x^2+y^2+2xy\cos2\gamma)\leq0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/395231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$ Can this integral be solved with contour integral or by some application of residue theorem?
$$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$
It has two poles at $\pm i$ and branch point of $-1$ while the integral is to b... | \begin{align*} \int_{0}^{\infty} \frac{\log (x + 1)}{x^2 + 1} \, dx
&= \int_{0}^{1} \frac{\log (x + 1)}{x^2 + 1} \, dx + \int_{1}^{\infty} \frac{\log (x + 1)}{x^2 + 1} \, dx \\
&= \int_{0}^{1} \frac{\log (x + 1)}{x^2 + 1} \, dx + \int_{0}^{1} \frac{\log (x^{-1} + 1)}{x^2 + 1} \, dx \quad (x \mapsto x^{-1}) \\
&= 2 \int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/396170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 9,
"answer_id": 5
} |
Chinese remainder theorem issue Let's say I have the following equations:
$$x \equiv 2 \mod 3$$
$$x \equiv 7 \mod 10$$
$$x \equiv 10 \mod 11$$
$$x \equiv 1 \mod 7$$
And I need to find the smallest x for which all these equations are correct. So:
$N = 3\times10\times11\times7 = 2310$
$N_1 = \frac{2310}{3} = 770$
$N_2 = ... | More simply: $ $ mod $3,11\!:\ x\equiv -1,\ $ so $ $ mod $33\!:\ x\equiv -1\ $ so $\ x = -1 + 33j.$
mod $7\!:\ 1 \equiv x \equiv -1 + 33j\ \Rightarrow\ j\equiv \frac{2}{33} \equiv \frac{2}{{-}2} \equiv -1\ $ so $\ j = -1 + 7k.$
Substituting: $\ x = -1 + 33j = -1 + 33(-1 + 7k) = -34 + 231k.$
mod $10\!:\ 7 \equiv x \e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/396505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A simple 2 grade equations system If we have:
$$x^2 + xy + y^2 = 25 $$
$$x^2 + xz + z^2 = 49 $$
$$y^2 + yz + z^2 = 64 $$
How do we calculate $$x + y + z$$
| $$x^2 + xy + y^2 = 25 \dots (1)$$
$$x^2 + xz + z^2 = 49 \dots(2)$$
$$y^2 + yz + z^2 = 64 \dots(3)$$
$(2)-(1)$
$x(z-y)+(z+y)(z-y)=24$
$\Rightarrow (x+y+z)(z-y)=24$
Similarly we get ,
$(x+y+z)(y-x)=15$ by $(3)-(2)$
$(x+y+z)(z-x)=39$ by $(3)-(1)$
Clearly Let $1/\lambda=(x+y+z)\ne 0$
Then we have,
$(z-y)=24\lambda$
$(y-x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
How to use "results from partial fractions"? Let ${a_n}$ be a sequence whose corresponding power series $A(x)=\sum_{i\geq 0}a_ix^i$ satisfies
$$A(x)=\frac{6-x+5x^2}{1-3x^2-2x^3}$$
The denominator can be factored into $(1-2x)(1+x)^2$. Using results from partial fractions, it can be shown that there exists constants $C_1... | We describe what we do once we have the coefficients $C_1,C_2,C_3$. It all comes from the expansion
$$\frac{1}{1-t}=1+t+t^2+t^3+\cdots.\tag{$1$}$$
Putting $t=2x$, we find that $\frac{C_1}{1-2x}$ has expansion
$$C_1+2C_1 x+4C_1x^2+8C_1x^3+\cdots.$$
Putting $t=-x$, we can in a similar way get the power series expansion ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Being inside or outside of an ellipse Let $A$ be a point $A$ not belonging to an ellipse $E$. We say that $A$ lies inside
$E$ if every line passing trough $A$ intersects $E$. We say that $A$ lies otside $E$
if some line passing trough $A$ does not intersect $E$. Let $E$ be the ellipse
with semi-axes $a$ and $b$. Show t... | Case $a:$
Let the point be $P(h,k)$ and any arbitrary line passing through $P$ be $y=mx+c$
$\implies k=m\cdot h+c\iff c=k-m\cdot h$
Let us find the intersection of the line with the given ellipse
$${So,}\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\implies \dfrac{x^2}{a^2}+\dfrac{(mx+c)^2}{b^2}$$
$$\text{On simplification,}x^2(b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/400025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve this system of equation. $x^2-yz=a^2$
$y^2-zx=b^2$
$z^2-xy=c^2$
How to solve this equation for $x,y,z$. Use elementary methods to solve (elimination, substitution etc.).
Given answer is:$x=\pm\dfrac{a^4-b^2c^2}{\sqrt {a^6+b^6+c^6-3a^2b^2c^2}}\,$, $y=\pm\dfrac{b^4-a^2c^2}{\sqrt {a^6+b^6+c^6-3a^2b^2c^2}}\,$ ... | Observe that $$(a^2)^2-b^2\cdot c^2=(x^2-yz)^2-(y^2-zx)(z^2-xy)=x(x^3+y^3+z^3-3xyz)$$
Similarly, $$b^4-c^2a^2=y(x^3+y^3+z^3-3xyz)\text{ and }c^4-a^2b^2=z(x^3+y^3+z^3-3xyz)$$
So, $$\frac x{a^4-b^2c^2}=\frac y{b^4-c^2a^2}=\frac z{c^4-a^2b^2}=\frac1{x^3+y^3+z^3-3xyz}=k\text{(say)}$$
Now, $x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/401436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Calculate the limit of two interrelated sequences? I'm given two sequences:
$$a_{n+1}=\frac{1+a_n+a_nb_n}{b_n},b_{n+1}=\frac{1+b_n+a_nb_n}{a_n}$$
as well as an initial condition $a_1=1$, $b_1=2$, and am told to find: $\displaystyle \lim_{n\to\infty}{a_n}$.
Given that I'm not even sure how to approach this problem, I tr... | It's obvious that $a_n$ and $b_n$ are in the same situation, so their limits highly depend on the initial values. Following are some points we can obtain from $a_1=1$ and $b_1=2$:
*
*$a_n>0$ and $b_n>0\ ;a_{n+1}-a_{n}=\frac{1+a_n}{b_n}>0$ and similarly $b_{n+1}-b_n>0$. Therefore, $\{a_n\}$ and $\{b_n\}$ are strictly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/401637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
Prove ${\frac {1+{a}^{3}}{1+a{b}^{2}}}+{\frac {1+{b}^{3}}{1+b{c}^{2}}}+{ \frac {1+{c}^{3}}{1+c{a}^{2}}}\ge 3 $
Let $a,b,c \ge0$, prove the on equality: $${\frac
{1+{a}^{3}}{1+a{b}^{2}}}+{\frac {1+{b}^{3}}{1+b{c}^{2}}}+{ \frac
{1+{c}^{3}}{1+c{a}^{2}}}\ge 3 $$
I tried: $$LHS = \sum\frac 1{1+ab^2}+\sum \frac {a^4}{a... | By AM-GM and Holder we obtain:
$$\sum_{cyc}\frac{1+a^3}{1+ab^2}\geq3\sqrt[3]{\frac{\prod\limits_{cyc}(1+a^3)}{\prod\limits_{cyc}(1+ab^2)}}=3\sqrt[3]{\frac{\sqrt[3]{\prod\limits_{cyc}(1+a^3)^3}}{\prod\limits_{cyc}(1+ab^2)}}=$$
$$=3\sqrt[3]{\frac{\sqrt[3]{\prod\limits_{cyc}(1+a^3)(1+b^3)^2}}{\prod\limits_{cyc}(1+ab^2)}}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/402904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Show that $n \ge \sqrt{n+1}+\sqrt{n}$ (how) Can I show that:
$n \ge \sqrt{n+1}+\sqrt{n}$ ?
It should be true for all $n \ge 5$.
Tried it via induction:
*
*$n=5$: $5 \ge \sqrt{5} + \sqrt{6} $ is true.
*$n\implies n+1$:
I need to show that $n+1 \ge \sqrt{n+1} + \sqrt{n+2}$
Starting with $n+1 \ge \sqrt{n} + \sqrt{n+1... | To add to your step, observe the following:
$$\sqrt{n}+1 = \sqrt{(\sqrt{n}+1)^2} = \sqrt{n+1+2\sqrt{n}} > \sqrt{n+1+1} = \sqrt{n+2}.$$
The "$>$" part comes from your assumption $n \ge 5$, so $2\sqrt{n} \ge 2\sqrt{5} > 1$. Now, we have:
$$n+1 \ge \sqrt{n} + \sqrt{n+1} + 1 = \sqrt{n+1} + (\sqrt{n}+1) > \sqrt{n+1} + \sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 1
} |
Show that $x^4+x^3+x^2+x+1$ and $x^4+2x^3+2x^2+2x+5$ cannot be a square when $x\neq3$ and $x\neq2$ respectively. As the title says, I need help showing that $x^4+x^3+x^2+x+1$ and $x^4+2x^3+2x^2+2x+5$ cannot be a square when $x\neq3$ and $x\neq2$ respectively, where $x$ is a natural number.
In order to do this, I hav... | Hint: IF either of these numbers were squares, then you could express them as $N^2$, where $N$ is some number. By what you know, you can write $N=c_0+c_1x+c_2x^2+\ldots+c_mx^m$ uniquely. Notice that $N$ uses at most three powers of $x$: $N=c_0+c_1x+c_2x^2$, since when you square it you need an $x^4$ to appear as the hi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/403473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to show that $\sqrt{1+\sqrt{2+\sqrt{3+\cdots\sqrt{2006}}}}<2$ $\sqrt{1+\sqrt{2+\sqrt{3+\cdots\sqrt{2006}}}}<2$.
I struggled on it, but I didn't find any pattern to solve it.
| $$\begin{aligned}\sqrt{1+\sqrt{2+\sqrt{3+\cdots \sqrt{n}}}}&<\sqrt{1+\sqrt{2+\sqrt{2^2+\cdots \sqrt{2^{2^{n-1}}}}}}\\&<\sqrt{1+\sqrt{2+\sqrt{2^2+\cdots }}}\\&=\sqrt{1+\sqrt{2}\cdot\sqrt{1+\sqrt{1+\cdots }}}\\&=\sqrt{1+\sqrt{2}\phi}\\&<2\end{aligned}$$
We can get a tighter bound for the limit by breaking the pattern a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/404653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
$1+1+1+1+1+1+1+1+1+1+1+1 \cdot 0+1 = 12$ or $1$? Does the expression $1+1+1+1+1+1+1+1+1+1+1+1 \cdot 0+1$ equal $1$ or $12$ ?
I thought it would be 12 this as per pemdas rule:
$$(1+1+1+1+1+1+1+1+1+1+1+1)\cdot (0+1) = 12 \cdot 1 = 12$$
Wanted to confirm the right answer from you guys. Thanks for your help.
| $1+1+1+1+1+1+1+1+1+1+1+1⋅0+1$
So, $1*0=0$ witch means that we Are Left with
$1+1+1+1+1+1+1+1+1+1+1+1(+0)$ witch is $12$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/405543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Prove inequality: $\sqrt {(x^2y+ y^2z+z^2x)(y^2x+z^2y+x^2z)} \ge \sqrt[3]{xyz(x^2+yz)(y^2+xz)(z^2+xy)} + xyz$
Let $x,y,z\in \mathbb R^+$ prove that:
$$\sqrt {(x^2y+ y^2z+z^2x)(y^2x+z^2y+x^2z)} \ge xyz + \sqrt[3]{xyz(x^2+yz)(y^2+xz)(z^2+xy)}$$
The inequality $\sqrt {(x^2y+ y^2z+z^2x)(y^2x+z^2y+x^2z)} \overset{C-S}{\g... | $$\sqrt {(x^2y+ y^2z+z^2x)(y^2x+z^2y+x^2z)} \overset{AM-GM}{\ge} 3xyz$$
DOES help, since you can also prove:
$$2\sqrt {(x^2y+ y^2z+z^2x)(y^2x+z^2y+x^2z)} \geq 3 \sqrt[3]{xyz(x^2+yz)(y^2+xz)(z^2+xy)}$$
Note that this is the same as
$$2^6(x^2y+ y^2z+z^2x)^3(y^2x+z^2y+x^2z)^3 \geq 3^6x^2y^2z^2(x^2+yz)^2(y^2+xz)^2(z^2+xy)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/406779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
How to solve integrals of type $ \int\frac{1}{(a+b\sin x)^4}dx$ and $\int\frac{1}{(a+b\cos x)^4}dx$ $$\displaystyle \int\frac{1}{(a+b\sin x)^4}dx,~~~~\text{and}~~~~\displaystyle \int\frac{1}{(a+b\cos x)^4}dx,$$
although i have tried using Trg. substution. but nothing get
| Use $\sin(x)= \frac{2\tan\frac{x}{2}}{1 +\tan^2 \frac {x}{2}}$ and $\cos(x)= \frac{1-\tan ^{2}\frac{x}{2}}{1 +\tan^{2}\frac {x}{2}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/410841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Limits problem to find the values of constants - a and b If $\lim_{x \to \infty}(1+\frac{a}{x}+\frac{b}{x^2})^{2x}=e^2$ Find the value of $a$ and $b$. Problem :
If $\lim\limits_{x \to \infty}\left(1+\frac{a}{x}+\frac{b}{x^2}\right)^{2x}=e^2$ Find the value of $a$ and $b$.
Please suggest how to proceed this problem :
... | Take logs of both sides to get
$$2 = \lim_{x \to \infty} 2 x \log{\left ( 1+ \frac{a}{x} + \frac{b}{x^2}\right)}$$
Use $\log{(1+y)} \sim y - (y^2)/2$ as $y \to 0$ to get
$$2 = \lim_{x \to \infty} \left ( 2 a + \frac{2 b-a^2}{x}\right)$$
So we require $a=1$. The value of $b$ is unimportant.
ADDENDUM
Details of the expa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/412996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that for every $n\in \mathbb{N}^{+}$, there exist a unique $x_{n}\in[\frac{2}{3},1]$ such that $f_{n}(x_{n})=0$ Let $f_{n}(x)=-1+x+\dfrac{x^2}{2^2}+\dfrac{x^3}{3^2}+\cdots+\dfrac{x^n}{n^2}$,
(1) Prove that for every $n\in \mathbb{N}^{+}$, then there exist unique $x_{n}\in[\frac{2}{3},1]$ such that
$f_{n}(x_{n})=0... | The $f_n$ converge to $f(x) = f_\infty(x) = -1 + \displaystyle \sum_{n=1}^{\infty} \frac{x^k}{k^2} $ = Li_2$(x)-1$ (the dilogarithm, minus $1$) with remainder terms $R_n(x) = \displaystyle \sum_{k=n}^{\infty} \frac{x^k}{k^2}$. Then:
*
*$x_n$ is a decreasing sequence that converges to $A \in (0,1)$ with $f(A)=0$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/413702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 2
} |
Evaluating $\lim_{n \to \infty} (1 + 1/n)^{n}$ I was recently thinking about how I could evaluate the famous limit of 'e' as I haven't ever seen a proof. I can't really find anything online so I've tried to evaluate the limit myself. And I was also thinking it would be nonsensical to use L'Hopital's rule, am I right?
S... | Here is the "standard" proof:
Let $a_{n}= \left(1 + \frac{1}{n} \right)^{n}= \frac{(n+1)^n}{n^n}$.
Then
$$\frac{a_{n+1}}{a_n}=\frac{(n+2)^{n+1}}{(n+1)^{n+1}}\frac{n^n}{(n+1)^n}=\frac{(n+2)^{n+1}n^{n+1}}{(n+1)^{2n+2}}\frac{n+1}{n}$$
$$=\left( \frac{n^2+2n}{(n+1)^{2}}\right)^{n+1}\frac{n+1}{n}=\left( 1-\frac{1}{(n+1)^{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?
Let $f(\frac ab)=ab$, where $\frac ab$ is irreducible, in $\mathbb Q^+$. What is $\sum_{x\in\mathbb Q^+}\frac 1{f(x)^2}$?
Club challenge problem. I don't think it's possible to do with only high s... | Note first that $f$ is multiplicative over the rationals in some sense. More precisely, if $\frac{a}{b}$ and $\frac{c}{d}$ are in reduced form and $(a,d) = (b,c) = 1$, we have that $f(\frac{ac}{bd}) = f(\frac{a}{b}) \cdot f(\frac{c}{d})$.
You have to justify this next step, but (if the sum behaves nicely) we can thus ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Find maximum value of $f(x)=2\cos 2x + 4 \sin x$ where $0 < x <\pi$ Find the maximum value of $f(x)$ where
\begin{equation}
f(x)=2\cos 2x + 4 \sin x \ \
\text{for} \ \ 0<x<\pi
\end{equation}
| $f(x)=2\cos 2x+4\sin x$
$=2(1-2\sin^2x)+4\sin x$
$=2-4\sin^2x+4\sin x$
$=2-(4\sin^2 x-2(2\sin x)+1)+1$
$=2+1-(2\sin x-1)^2\le (2+1)=3$
Equality holds iff $2\sin x-1=0\Rightarrow \sin x=1/2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/419150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Proving there are no integer solutions for $3x^2=9+y^3$
Prove there are no $x,y\in\mathbb{Z}$ such that $3x^2=9+y^3$.
Initial proof
Let us assume there are $x,y\in\mathbb{Z}$ that satisfy the equation, which can be rewritten as $$3(x^2-3)=y^3.$$ So, $$3 \mid y \Rightarrow 3^3 \mid y^3 \Rightarrow 3^2 \mid x^2 - 3.$$ ... | Assume that the expression has integer solutions, or atleast one for that matter:
Work in modulus $3$, if:
$$3(x^2-3)=y^3$$
$y^3$ must be a multiple of $3$. Then we can set $y=3k$ where $k$ is an integer. Now we have:
$$27k^3=3(x^2-3)$$
Dividing both sides by $3$ gives us $9k^3=x^2-3$
Taking $-3$ to the other side ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/419653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Decimal pattern in division of two digit numbers by 9 Can some one explain how this is possible ?
1) 13 / 9 = 1.(1 + 3) = 1.444 ...
2) 23 / 9 = 2.(2 + 3) = 2.555 ...
3) 35 / 9 = 3.(3 + 5) = 3.888 ...
4) 47 / 9 = 4.(4 + 7) = 4.(11) → 4.(11 - 9) = 5.222 ...
5) 63 / 9 = 6.(6 + 3) = 6.(9) → 6.(9 -... | Let $a$ be the tens digit and $b$ be the ones digit so that our two digit number is $10a+b$. In general, the pattern seems to be:
$$
\dfrac{10a+b}{9} = (a+j).kkk...
$$
where:
$$(j,k)=
\begin{cases}
(0,a+b) & \text{if }a+b \in \{0,1,2,...,8\} \\
(1,a+b-9) & \text{if }a+b \in \{9,10,11,...,17\} \\
(2,a+b-18) & \text{if }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/419911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find domain of $\log_4(\log_5(\log_3(18x-x^2-77)))$
Problem:$\log_4(\log_5(\log_3(18x-x^2-77)))$
Solution:
$\log_3(18x-x^2-77)$ is defined for $(18x-x^2-77) \ge 3$
$(x^2-18x+77) \le -3$
$(x^2-18x+80) < 0$ {As it can't be 0}
$(x-8)(x-10)<0$
$8<x<10$
So domain is $(8,10)$
Am I doing right ??
| HINT:
For for real $m>0$ $\log_ma,$ is defined if $a>0$ as if $\log_ma=b\iff a=m^b>0$
$\log_4(\log_5(\log_3(18x-x^2-77)))$ is defined if $\log_5(\log_3(18x-x^2-77))>0$
$\implies \log_3(18x-x^2-77)>5^0=1$
$\implies (18x-x^2-77)>3^1=3$
$\implies x^2-18x+80<0$
Now, the roots of $x^2-18x+80=0$ are $\frac{18\pm\sqrt{18^2-4\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/420769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A question on geometry? I wanted to know, given quadrilateral ABCD such that $AB^2+CD^2=BC^2+AD^2$ , prove that $AC⊥BD$ .
Help.
Thanks.
|
Here is Geometry method:
$AB^2=BE^2+AE^2, DC^2=DF^2+CF^2, BC^2=BE^2+CE^2,AD^2=AF^2+DF^2$
$AB^2+CD^2=BC^2+AD^2 \implies BE^2+AE^2+DF^2+CF^2=BE^2+CE^2+AF^2+DF^2$
$\implies AE^2+CF^2=CE^2+AF^2 \implies AE^2- AF^2=CE^2-CF^2$ .....<1>
$AE \ge AF, CE \le CF $ so for <1>:
LHS $\ge 0$, RHS $ \le 0$, the only condition is both... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/421844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
$a+b+c =0$; $a^2+b^2+c^2=1$. Prove that: $a^2 b^2 c^2 \le \frac{1}{54}$ If a,b,c are real numbers satisfying $a+b+c =0; a^2+b^2+c^2=1$.
Prove that $a^2 b^2 c^2 \le \frac{1}{54}$.
| As several have shown above, a, b and c are roots of the cubic $abc=t^3-.5t$. In order for this to have three real roots, abc must lie between the maximum and minimum of $t^3-.5t$. At one of these extrema, two of the roots are equal to the location of the extrema, and the third is found from the requirement that the su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 6
} |
Sum of greatest common divisors As usually, let $\gcd(a,b)$ be the greatest common divisor of integer numbers $a$ and $b$.
What is the asymptotics of
$$\frac{1}{n^2} \sum_{i=1}^{i=n} \sum_{j=1}^{j=n} \gcd(i,j)$$
as $n \to \infty?$
| We can rewrite the sum as
$$
\begin{align}
S(n)&=\frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^n\gcd(i,j) \\ &= \frac{1}{n^2}\sum_{g=1}^n\sum_{i\le\lfloor n/g\rfloor}\sum_{\substack{j\le\lfloor n/g\rfloor\\(i,j)=1}} g \\
&= \frac{1}{n^2}\sum_{g=1}^n g\left(-1+2\sum_{i=1}^{\lfloor n/g\rfloor} \varphi(i)\right) \\
&= -\frac{n(n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. Can anyone give me a hint to evaluate this integral?
$$\int_0^\infty \frac{dx}{1+x^4}$$
I know it will involve the gamma function, but how?
| You could mess about with integration by substitution methods.
For instance, if you change the expression to this:
$$\frac{1}{2}\int_0^\infty \frac{(x^2+1)-(x^2-1)}{x^4+1}dx = \frac{1}{2}\int_0^\infty \frac{(x^2+1)}{x^4 + 1}-\frac{(x^2-1)}{x^4+1}dx$$
which you can then separate out as:
$$\frac{1}{2}\left[\int_0^\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 3
} |
Why $x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$?
Why $f(x) = x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$ (where $\zeta = \zeta_7$ is a primitive root of the unit) over $\mathbb{Q}$?
Of course it's irreducible by the Eisenstein criterion, however it apparent... | Compute:
$$\begin{align*}
(1+2(\zeta+\zeta^2+\zeta^4))^2+7&=\bigg[1^2+4(\zeta+\zeta^2+\zeta^4)+4(\zeta+\zeta^2+\zeta^4)^2\bigg]+7\\[0.1in]
&=\bigg[1+4(\zeta+\zeta^2+\zeta^4)+4(\zeta^2+\zeta^4+\zeta^8+2\zeta^3+2\zeta^5+2\zeta^6)\bigg]+7\\[0.1in]
(\mathsf{\text{because }}\zeta^8=\zeta)\quad&=\bigg[1+4(\zeta+\zeta^2+\zeta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$ We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$
I have tried and it gets confusing.
| These kind of questions often benefit from the identity $a^2-b^2=(a-b)(a+b)$ in conjunction with Pythagorean trig identitities. Here,
$$
\begin{align}
\frac{\sin t-\cos t+1}{\sin t+\cos t -1}&=\frac{(\sin t+1)-\cos t}{(\sin t+\cos t) -1}\cdot\frac{(\sin t+1)+\cos t}{(\sin t+\cos t) +1}\\
&=\frac{\sin^2 t+2\sin t+1-\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 1
} |
Prove $\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A$ by the Pythagorean theorem. How do I use the Pythagorean Theorem to prove that $$\frac{\cos^2 A}{1 - \sin A} = 1 + \sin A?$$
| Let us denote base of right angle triangle as b, perpendicular ( height ) as p, and hypotenuse as h,
$$\cos A = \frac{b}{h} ; \qquad \sin A = \frac{p}{h}\tag{i}$$
Therefore, $$\frac{\cos^2A}{1-\sin A} = \frac{\frac{b^2}{h^2}}{1-\frac{p}{h}}$$
[By putting the values of $\cos A$ and $\sin A$ from $(i)$]
Which after simp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
How to find the partial derivative of this complicated function? Find the partial derivative with respect to a (treat other variables as constants):
$\displaystyle f(a,b,c)=\frac {a}{\sqrt{a^2+8bc}}-\frac {a^r}{a^r+b^r+c^r}$
The article I'm reading says it should be:
$\displaystyle \frac {\sqrt{a^2+8bc}-\dfrac{a^2}{\sq... | Form the partial derivatives quotient rule, we know:
$$\displaystyle \frac{\partial}{\partial a} \left(\frac{g}{h}\right) = \frac{g' h - g h'}{h^2}$$
Given the function:
$$\displaystyle f(a,b,c)=\frac {a}{\sqrt{a^2+8bc}}-\frac {a^r}{a^r+b^r+c^r}$$
Lets do each term separately, so we have:
$$\frac{(1) \sqrt{a^2+8bc}-(a)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/428742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A proposed proof by induction of $1+2+\ldots+n=\frac{n(n+1)}{2}$
Prove: $\displaystyle 1+2+\ldots+n=\frac{n(n+1)}{2}$.
Proof
When $n=1,1=\displaystyle \frac{1(1+1)}{2}$,equality holds.
Suppose when $n=k$, we have $1+2+\dots+k=\frac{k(k+1)}{2}$
When $n = k + 1$:
\begin{align}
1+2+\ldots+k+(k+1) &=\frac{k(k+1)}{2}+k+1 ... | Nice work! If you want to take a bit more time, you can note that $$\frac{k^2+3k+2}2=\frac{k^2+2k+k+2}2=\frac{k(k+2)+1(k+2)}2=\frac{(k+1)(k+2)}2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/429931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How can I get polynomial $p(x)$? $p(x)$ is divided evenly into $x^{2}+1$, and $p(x)+1$ is divided evenly into $x^{3}+x^{2}+1$. How can I get $p(x)$?
| The given data tells us:
$$\begin{align}r(x)(x^2 + 1) &= p(x)\\
s(x)(x^3 + x^2 + 1) &= p(x) + 1
\end{align}$$
The above equations imply that
$$s(x)(x^3 + x^2 + 1) - r(x)(x^2 + 1) = 1. \tag1$$
Does the structure of the equation ring any bells? Yes! Bezout's identity!
Now use Euclid's division algorithm on $x^2 + 1$ and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}>2$ for $x > 0$
Let $x>0$. Show that
$$(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}>2.$$
Do you have any nice method?
My idea $F(x)=(x+1)^{\frac{1}{x+1}}+x^{-\frac{1}{x}}$
then we hvae $F'(x)=\cdots$ But it's ugly.
can you have nice methods? Thank you
by this I have see thi... | Remark: Bernoulli inequality and AM-GM are enough.
Letting $x = \frac{1}{y}$, it suffices to prove that, for all $y > 0$,
$$\left(1 + \frac{1}{y}\right)^{\frac{y}{1+y}} + y^y > 2.$$
If $y \ge 3$, clearly the inequality is true.
If $0 < y < 3$, we have
\begin{align*}
\left(1 + \frac{1}{y}\right)^{\frac{y}{1+y}} + y^y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/434174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to integrate these integrals $$\int^{\frac {\pi}2}_0 \frac {dx}{1+ \cos x}$$
$$\int^{\frac {\pi}2}_0 \frac {dx}{1+ \sin x}$$
It seems that substitutions make things worse:
$$\int \frac {dx}{1+ \cos x} ; t = 1 + \cos x; dt = -\sin x dx ; \sin x = \sqrt{1 - \cos^2 x} = \sqrt{1 - (t-1)^2} $$
$$ \Rightarrow
\int \frac ... | $$
\begin{array}{lcl}
\int_0^{\pi/2} \frac{1}{1+\cos x}dx &=& \int_0^{\pi/2} \frac{1}{(\cos^2(x/2)+\sin^2(x/2))+(\cos^2 (x/2)-\sin^2(x/2))}dx\\
&=& \int_0^{\pi/2} \frac{1}{2\cos^2(x/2)}dx\\
&=& \int_0^{\pi/4}\frac{1}{\cos^2u}du=\tan\frac{\pi}{4}=1
\end{array}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/434247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
} |
How find this value $\frac{a^2+b^2-c^2}{2ab}+\frac{a^2+c^2-b^2}{2ac}+\frac{b^2+c^2-a^2}{2bc}$ let $a,b,c$ such that
$$\left(\dfrac{a^2+b^2-c^2}{2ab}\right)^2+\left(\dfrac{b^2+c^2-a^2}{2bc}\right)^2+\left(\dfrac{a^2+c^2-b^2}{2ac}\right)^2=3,$$
find the value
$$\dfrac{a^2+b^2-c^2}{2ab}+\dfrac{a^2+c^2-b^2}{2ac}+\dfrac{b^... | I think you might be talking about a triangle, with $a$, $b$, $c$ as it's sides of a triangle. If the angles opposite to side a, b, c are A, B,C, then $$\cos A=\frac{b^2+c^2-a^2}{2cb}$$ and similarly, $$\cos B=\frac{a^2+c^2-b^2}{2ac}$$ and $$\cos C=\frac{b^2+a^2-c^2}{2ab}$$ If it is true then $\cos A = \cos B = \cos C ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
If $\sin a+\sin b=2$, then show that $\sin(a+b)=0$ If $\sin a+\sin b=2$, then show that $\sin(a+b)=0$.
I have tried to solve this problem in the following way :
\begin{align}&\sin a + \sin b=2 \\
\Rightarrow &2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)=2\\
\Rightarrow &\sin\left(\frac{a+b}{2}\rig... | The maximum value of the $sin$ function is 1, at $\pi / 2 + 2k\pi$, for any integer $k$.
So $a = \pi / 2 + 2k\pi$, and $b = \pi / 2 + 2l\pi$, for any integers $k$, $l$.
Therefore, $\sin(a + b) = \sin(\pi + 2(k + l)\pi) = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/435513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Why $\sum_{k=1}^n \frac{1}{2k+1}$ is not an integer? Let $S=\sum_{k=1}^n \frac{1}{2k+1}$, how can we prove with elementary math reasoning that $S$ is not an integer?
Can somebody help?
| Let $p=2k'+1$ be the largest odd prime number less than or equal to $2n+1$. Now consider the sum:
$$\frac13 + \frac15 + \frac17 + \cdots + \frac{1}{2k'-1} + \frac{1}{p} + \frac{1}{2k'+3} + \cdots + \frac{1}{2n+1}.$$
We can find a common denominator for this fraction by taking the product:
$$3\cdot5\cdots (2k'-1) \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
What is the integral of $\int e^x\,\sin x\,\,dx$? I'm trying to solve the integral of $\left(\int e^x\,\sin x\,\,dx\right)$ (My solution):
$\int e^x\sin\left(x\right)\,\,dx=$
$\int \sin\left(x\right) \,e^x\,\,dx=$
$\left(\sin(x)\,\int e^x\right)-\left(\int\sin^{'}(x)\,\left(\int e^x\right)\right)$
$\left(\sin(x)\,e^x\r... | Another way via leibniz theorem,
$$ \frac{d^4 (e^x \sin x)}{dx^4} =e^x \left[\binom{4}{0} \sin x+ \binom{4}{1} \cos x - \binom{4}{2} \sin x - \binom{4}{3} \cos x+ \binom{4}{4} \sin x\right]$$
Or,
$$ \frac{1}{ \sum_{k=0}^2 \binom{4}{2k} } \frac{d^4 (e^x \sin x)}{dx^4} = e^x \sin x$$
Integrate both sides and apply leibn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
Given that $\cos x =-3/4$ and $90^\circGiven that $\;\cos x =-\frac{3}{4}\,$ and $\,90^\circ<x<180^\circ,\,$ find $\,\tan x\,$ and $\,\csc x.$
This question is quite unusual from the rest of the questions in the chapter, can someone please explain how this question is solved? I tried Pythagorean Theorem, but no luck. I... | The cosine of an angle corresponds to the $x$-coordinate in the unit circle, and the sine of an angle cooresponds to its $y$-coordinate on the unit circle.
Note that $\,\cos = -\frac 34 < 0\,$ if and only if the angle $x$ terminates in either the second or third quadrant, where the angle $x$ is measured with respect to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/439369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Recursion Question - Trying to understand the concept Just trying to grasp this concept and was hoping someone could help me a bit. I am taking a discrete math class. Can someone please explain this equation to me a bit?
$f(0) = 3$
$f(n+1) = 2f(n) + 3$
$f(1) = 2f(0) + 3 = 2 \cdot 3 + 3 = 9$
$f(2) = 2f(1) + 3 = 2 \cdo... | Simply use substitution.
We are given the initial value $$\color{blue}{\bf f(0) = 3}\tag{given}$$
Each subsequent value of the function $f$ depends on the preceding value. So the function evaluated at $(n + 1)$ depends (is defined, in part) on the function's value at $n$: That's what's meant by a recursive definition o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/441718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Proof that $ \lim_{x \to \infty} x \cdot \log(\frac{x+1}{x+10})$ is $-9$ Given this limit:
$$ \lim_{x \to \infty} x \cdot \log\left(\frac{x+1}{x+10}\right) $$
I may use this trick:
$$ \frac{x+1}{x+1} = \frac{x+1}{x} \cdot \frac{x}{x+10} $$
So I will have:
$$ \lim_{x \to \infty} x \cdot \left(\log\left(\frac{x+1}{... | $$
\begin{align}
\lim_{x\to\infty}x\log\left(\frac{x+1}{x+10}\right)
&=\lim_{x\to\infty}x\log\left(1-\frac{9}{x+10}\right)\\
&=\lim_{x\to\infty}\frac{\log\left(1-\frac{9}{x+10}\right)}{-\frac{9}{x+10}}\left(-\frac{9x}{x+10}\right)\\[9pt]
&=\lim_{u\to0}\frac{\log(1+u)}{u}\lim_{x\to\infty}\left(-\frac{9x}{x+10}\right)\\[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/442254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Finding perpendicular bisector of the line segement joining $ (-1,4)\;\text{and}\;(3,-2)$
Find the perpendicular bisector of the line joining the points $(-1,4)\;\text{and}\;(3,-2).\;$
I know this is a very easy question, and the answer is an equation. So any hints would be very nice. thanks
| If the point $P=(x,y)$ lies on the perpendicular bisector of the points $A=(-1,4)$ and
$B= (3, -2)$, then the distances $PA$ and $PB$ must be the same.
Then
$$
PA^2 = PB^2 \\
\Rightarrow (x+1)^2 + (y-4)^2 = (x-3)^2 + (y+2)^2 \\
\Rightarrow x^2 +2x + 1 + y^2 -8y +16 = x^2 -6x +9 +y^2 + 4y + 4 \\
\Rightarrow 2x - 8y +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/442781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Discrete Math Question: arithmetic progression A lumberjack has $4n + 110$ logs in a pile consisting of n layers. Each layer has two more logs than the layer directly above it. If the top layer has six logs, how many layers are there? Write the steps to calculate the equation for the problem and state the number of lay... | $$6+8+10+\cdots+(6+2n-2)=6n+(0+2+4+\cdots+2n-2)=\\6n+2(0+1+2+\cdots+n-1)=6n+2\frac{(n-1)n}{2}=6n+(n-1)n=n^2+5n$$
You were told that this equals $4n+110$, so $n^2+5n=4n+110$, which is a quadratic equation $n^2+n-110=(n+11)(n-10)=0$. The only positive solution is $n=10$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/443079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Calculating $\lim_{x\to 0}\left(\frac{1}{\sqrt x}-\frac{1}{\sqrt{\log(x+1)}}\right)$ Find the limit
$$\lim_{x\to 0}\left(\frac{1}{\sqrt x}-\frac{1}{\sqrt{\log(x+1)}}\right)$$
| Hint
$$\frac{\log(1+x)}{x}\to 1$$
$$\frac{\log(1+x)-x}{x^2}\to -\frac{1}{2}$$
Further Hint
$$\displaylines{
\frac{1}{{\sqrt x }} - \frac{1}{{\sqrt {\log (x + 1)} }} = \frac{{\sqrt {\log (x + 1)} - \sqrt x }}{{\sqrt {x\log \left( {1 + x} \right)} }}\left( {\frac{{\sqrt {\log (x + 1)} + \sqrt x }}{{\sqrt {\log (x + ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/444404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$? Are there solutions in integers $a,b>1$ to the following simultaneous congruences?
$$
a^4\equiv 1 \pmod{b^2} \quad \mathrm{and} \quad b^4\equiv1 \pmod{a^2}
$$
A brute-force search didn't turn up any small ones, but I also don't see how to rule... | Maybe like this:
First observe that b^2 | (a^4-1) and a^2 | (b^4-1). Hence, we can write a^4-1=kb^2 and b^4-1=ma^2, with k, m being positive integers.
So that gives then that a^4-1 = ((b^4-1)/m)^2-1=(b^8-2b^4+1)/m^2-1 = kb^2
That means that b^8-2b^4+1 = kb^2*m^2+m^2
Hence b^2(b^6-2b^2-km^2)=m^2-1
Now note that b^2 the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/446269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 0
} |
Evaluating $\lim\limits_{x \to 0} \frac1{1-\cos (x^2)}\sum\limits_{n=4}^{\infty} n^5x^n$ I'm trying to solve this limit but I'm not sure how to do it.
$$\lim_{x \to 0} \frac1{1-\cos(x^2)}\sum_{n=4}^{\infty} n^5x^n$$
I thought of finding the function that represents the sum but I had a hard time finding it.
I'd apprecia... | Of course finding a formula for the sum of the series is not needed to solve this.
$$\begin{align}
\cos(x^2) &= 1 - \frac{x^4}{2} + O(x^8)\qquad\text{as }x \to 0
\\
\frac{1}{1-\cos(x^2)} &= 2 x^{-4} + O(1)
\\
\sum_{n=4}^\infty n^5 x^n &= 4^5 x^4 + O(x^5)
\\
\frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty n^5 x^n &= 2\cdot 4^5 +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/448100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
convert ceil to floor Mathematically, why is this true?
$$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a+b-1}{b}\right\rfloor$$
Assume $a$ and $b$ are positive integers.
Is this also true if $a$ and $b$ are real numbers?
| This is not true in general, e.g. take $b = 1/2$ and $a = 2,$ so the LHS is $4$ while the RHS is $3.$
Suppose $b\nmid a,$ since that case is trivial. Use the division algorithm to write $a = qb + r,$ where $q \in \mathbb{N}\cup \{0\}$ and $0 < r < b.$ Then the LHS is $q+1$ while the RHS is $\lfloor (q+1) + \frac{r-1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/448300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 0
} |
Prove that $2^n < \binom{2n}{n} < 2^{2n}$ Prove that $2^n < \binom{2n}{n} < 2^{2n}$. This is proven easily enough by splitting it up into two parts and then proving each part by induction.
First part: $2^n < \binom{2n}{n}$. The base $n = 1$ is trivial. Assume inductively that some $k$ satisfies our statement. The indu... | The central binomial coefficient is largest implies a lower bound of $(2^{2n})/(2n+1)$. This exceeds $2^n$ for $n>1$ and is almost its square for large $n$.
There are more subsets than subsets of a fixed cardinality implies an upper bound of $2^{2n}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/448861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 7,
"answer_id": 6
} |
Is $x^2-y^2=1$ Merely $\frac 1x$ Rotated -$45^\circ$? Comparing their graphs and definitions of hyperbolic angles seems to suggest so aside from the $\sqrt{2}$ factor:
and:
| Write it in polar coordinates:
$$x^2 - y^2 = 1 \ \to \ r^2(\cos^2 \theta - \sin^2 \theta) r^2 = r^2\cos(2\theta) = 1$$
$$xy=1 \ \to \ r^2\sin\theta\cos\theta=\frac{1}{2}r^2\sin(2\theta) =1$$
Or:
$$\frac{1}{2}r^2\sin(2\theta)=\frac{1}{2}r^2\cos(2(\theta + 45^\circ))=\left(\frac{r}{\sqrt{2}}\right)^2\cos(2(\theta + 45^\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/448961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 0
} |
2 Different integrals of $\int \left ( \tan{x}\right ) ^3 dx $. My friend asked me why this function has 2 different integrals. I'm very confused.
\begin{align}
\int \left ( \tan{x}\right ) ^3 dx &=\int \left ( \tan{x} \right )^2 \tan{x}dx \\
&=\int \left ( \sec^2 {x} -1 \right ) \tan{x} dx \\
&=\int \tan{x} \left ( \t... | Note that $\frac{1}{2}\sec^2 x$ and $\frac{1}{2}\tan^2 x$ differ by a constant. Since there is always an arbitrary constant of integration, the two answers are the same answer.
In the same way $x^2+17+C$ and $x^2+C$ are both correct integrals of $2x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/449290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Simplifying compound fraction: $\frac{3}{\sqrt{5}/5}$ I'm trying to simplify the following:
$$\frac{3}{\ \frac{\sqrt{5}}{5} \ }.$$
I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I multiply the numerator by the reciprocal of the denominator.
The pr... | \begin{align*}\frac{3}{\frac{\sqrt{5}}{5}} &= 3 \cdot \frac{5}{\sqrt 5}\\
&= 3 \cdot \frac{5}{\sqrt 5} \cdot 1\\
&= 3 \cdot \frac{5}{\sqrt 5} \cdot \frac{\sqrt 5}{\sqrt 5}\\
&= 3 \cdot \frac{5\sqrt 5}{5}\\
&= 3\sqrt 5
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/450158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 0
} |
Interpreting the ; in a series This question is linked to this question.
So, suppose I set $n=5$. Given the following formula:
$$\frac{1}{n}, \dots , \frac{n-1}{n} $$
Am I suppose to get:
$$
\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \hspace{8.2cm}(1)
$$
Or
$$
\frac{1}{5}, \frac{1}{4}, \frac{1}{3}, \frac{2}... | You are supposed to get (1), the complete enumeration makes it clear. The semicolon is only put for clarity in separating terms with different denumerator. Skipping is implicit because the author is enumerating the values of a set, hence repeating them has no effect.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/450291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
divisibility test let $n=a_m 10^m+a_{m-1}10^{m-1}+\dots+a_2 10^2+a_1 10+a_0$, where $a_k$'s are integer and $0\le a_k\le 9$, $k=0,1,2,\dots,m$, be the decimal representation of $n$
let $S=a_0+\dots+a_m$, $T=a_0-a_1\dots+(-1)^ma_m$
then could any one tell me how and why on the basis of divisibility of $S$ and $T$ by $2,... | The presence of many subscripts can make something simple look not so simple. So we deal with a number like $N=a_4\cdot 10^4+a_3\cdot 10^3 +a_2\cdot 10^2 +a_1\cdot 10^1+a_0$, where the $a_i$ are digits. Let $S=a_4+a_3+a_2+a_1+a_0$.
We have
$$N-S=a_4\cdot 9999+a_3\cdot 999+a_2\cdot 99+a_1\cdot 9.\tag{1}$$
The right-ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.