Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Very hard inequality: $\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le k_p \sqrt{a+b+c}.$ Given $p>0$. Find the smallest real number $k_p$ such that the following inequality holds for any non-negative reals $a,b,c$:
$$\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} \le k_p \sqrt{... | Consider the vectors $v = \left(\sqrt a,\sqrt b,\sqrt c\right)$ and $u = \left(\frac{\sqrt a}{\sqrt{a + pb}},\frac{\sqrt b}{\sqrt{b + pc}},\frac{\sqrt c}{\sqrt{c + pa}}\right)$
Applying the scalar product inequality, we get:
$$\frac{a}{\sqrt{a+pb}}+\frac{b}{\sqrt{b+pc}}+\frac{c}{\sqrt{c+pa}} = v\cdot u \leq |v||u| = \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/735033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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LU Factorisation (4x4 matrix) - most efficient method I understand how to do LU factorisation but I'm not sure I'm being very efficient. I first find the row echelon form of A, noting the elementary operations $E_i$ in order.
$$
E_1E_2...E_nA = U $$ then $$ L = E_1^{-1}E_2^{-1}...E_n^{-1}
$$
But is this the quickest wa... | We can use Doolittle's Method:
$$\begin{bmatrix} 1 & 0 & 0 &0\\ l_{21} & 1 & 0 &0 \\ l_{31} & l_{32} & 1 &0 \\ l_{41} & l_{42} & l_{43} & 1 \end{bmatrix} \cdot \begin{bmatrix} u_{11} & u_{12} & u_{13} &u_{14}\\ 0 & u_{22} & u_{23} &u_{24} \\ 0 & 0 & u_{33} &u_{34} \\0 & 0 & 0 & u_{44} \end{bmatrix} = \begin{bmatrix} 1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/735388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is that function? What is the set of the convergence in the reals of the series
$$f(x):= \sum\limits_{n=1}^{\infty} \frac 1 n \sin \left (n+\frac x n\right)?$$ Is the function $f(x)$ bounded?
Edit: more exact title.
| For any fixed $x$; and any $N\geq 1$
\begin{align*}
\sum_{n=1}^{N} \frac{1}{n} \sin \left (n+\frac x n\right) &=
\sum_{n=1}^{N} \frac{1}{n} \sin n \cos \frac{x}{n} + \sum_{n=1}^{N} \frac{1}{n} \cos n \sin \frac{x}{n}
\end{align*}
Now,
*
*$\frac{\sin n}{n} \cos \frac{x}{n} = \frac{\sin n}{n} - \frac{\sin n}{2n^3}x^... | {
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"timestamp": "2023-03-29T00:00:00",
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How to solve $x$ in $(x+1)^4+(x-1)^4=16$? I'm trying my hand on these types of expressions. How to solve $x$ in $(x+1)^4+(x-1)^4=16$? Please write any idea you have, and try to keep it simple. Thanks.
| Let denote by $P(X)$ the following polynomial:
$$P(X)=(X+1)^4+(X-1)^4-16.\quad\quad(1)$$
We immediately see that $X=1$ and $X=-1$ are roots of $P(X)$. Therefore, $P(X)$ can be written as:
$$P(X)=(X-1)(X+1)Q(X), \quad\quad\;\;\;\;\;\;(2)$$
where $Q(X)$ is a polynomial with degree $2$. Hence, $Q(X)=aX^2+bX+c$ for some $... | {
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Transformation Matrix $M_B^B$ of $P_3$ for $B = (1,x,x^2,x^3)$. Is that correct? I have the following task and just wanted to check weather this is (written) correct(ly).
Let $V$ be the vector space of all polynomials of grade $\le 3$ and $f: V \rightarrow V, p \rightarrow p'$ an $\mathbb{R}-$linear map. Calculate the ... | Let $V$ be a vector space and $T$ an operator on $V$. Since for the basis $(v_1,...,v_n)$ of $V$, $T$ maps each $v_k$ as a linear combination $Tv_k =a_{1,k}v_1 \ + \ ...\ + \ a_{n,k}v_n$ and is represented by the matrix
$$\left[\begin{matrix}
a_{1,1}&\cdots& &a_{1,n} \\
\vdots& \ddots & &\vdots\\
a_{n,1}&\cdots & ... | {
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Show that $\int_0^1 \! \frac{1+x^2}{1+x^4} \, \operatorname d \!x=1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\cdots$ As the title states, trying to solve $$\int_0^1 \! \frac{1+x^2}{1+x^4} \, \operatorname d \!x=1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\cdots$$
| Simply express the integrand as the series that converges for $|x| < 1$ and then interchange the order of summation and integration.
Hint: It is straightforward to show that
$$\frac{1 + x^2}{1 + x^4} = 1 + x^2 - x^4 - x^6 + x^8 + x^{10} - \cdots$$
because
$$\frac{1 + x^2}{1 + x^4} = \frac{1}{1 + x^4} + \frac{x^2}{1 + x... | {
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"timestamp": "2023-03-29T00:00:00",
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How to find the coefficient of $x^8$ in $\prod\limits_{i=1}^{10}{\left(x-i\right)}$? How to find the coefficient of $x^8$ in $(x-1) (x-2) . . .(x-10)$.
Is there any general formula to solve this kind of problems?
| The roots of the polynomial $(x - 1)(x-2)\cdots(x-10) = 0$ are $x = 1,2,\dots,10$.
Apply Vieta's Formulas to deduce that the coefficient of $x^8$ is the sum of all pairwise products of these roots (a.k.a the second elementary symmetric function of the roots):
$$(1\cdot2 + 1\cdot 3 + \dots + 1\cdot 10) + (2\cdot 3 + 2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741690",
"timestamp": "2023-03-29T00:00:00",
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If $AD=BD$, $\angle ADC=3\angle CAB$, $AB=\sqrt{2}$, $BC=\sqrt{17}$, $CD=\sqrt{10}$. Find $AC$ In quadrilateral $ABCD$, we have
$$AD=BD,\angle ADC=3\angle CAB,AB=\sqrt{2},$$ $$BC=\sqrt{17},CD=\sqrt{10}$$
Find the $AC=?$
My idea: let $$\angle CAB=x.\angle ADC=3x,\angle ADB=y,$$
then we have
$$\angle CAD=90-\dfrac{y}{2}-... | You only have to solve the 4-equation system below:
\begin{cases}
\overline{AC}^2+\overline{AB}^2-2\overline{AC}\cdot \overline{AB}\cos(x)=\overline{BC}^2 \\
\overline{AD}^2+\overline{CD}^2-2\overline{AD}\cdot \overline{CD}\cos(3x)=\overline{AC}^2 \\
\overline{BD}^2+\overline{CD}^2-2\overline{BD}\cdot \overline{CD}\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/743667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integrating a function with substitution Totally forgot how to integrate.
$$ \int \frac{1}{x^2 \sqrt{x^2+4}}dx$$
Just need a tip, for this what would I use to substitute?
| Put $x = 2 \tan t $, then $dx = 2 \sec^2 t dt $. and $\sqrt{x^2 +4} = \sqrt{ 4 \tan^2 t + 4 } = 2 \sec t$ hence,
$$ \int \frac{dx}{x^2 \sqrt{x^2+4}} = \int \frac{2 \sec^2 t dt}{4 \tan^2 t 2 \sec t} = \frac{1}{4} \int \frac{ \sec t dt }{\tan^2 t} = \frac{1}{4} \int \frac{\frac{1}{\cos t}}{\frac{\sin^2t}{\cos^2 t}} = \fr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $\sigma(p^m)=2^n$ for prime $p$,then $m=1$ and $n$ is prime Exercise from Beginning Number Theory by Neville Robbins:
Let $\sigma(a)$ denote the sum of divisors of $a$.Then we have to prove that if $\sigma(p^m)=2^n$ for some prime $p$,then $m=1$ and $n$ is a prime.
The result does not hold for $p=2$.Now,$$\dfrac{p... | Excluding $m = 0$ - when $\sigma(p^0) = 2^0$ - we see that $p$ must be an odd prime, since $\sigma(2^m) \equiv 1 \pmod{2}$. So $\sigma(p^m) = 1 + p + \dotsc + p^m \equiv m+1 \pmod{2}$ forces $m$ to be odd. Then let $k = \frac{m+1}{2}$ in
$$\sigma(p^m) = \frac{p^{2k}-1}{p-1} = (p^k+1)\cdot\frac{p^k-1}{p-1}.$$
If $\sigma... | {
"language": "en",
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Find sequence function and general rule the function $$a_{n+2}=3a_{n+1}-2a_n+2$$
is given, and $$a_0=a_1=1, (a_n)_{n\ge0}$$
multiplying everything by $$/\sum_{n=0}^\infty x^{n+2}$$
also adding $$\sum_{n=0}^\infty (a_{n+2}x^{n+2}+a_1x+a_0)-a_1x-a_0=\sum_{n=0}^\infty (3a_{n+1}x^{n+2}+a_0)-a_0-\sum_{n=0}^\infty 2a_nx^{n+2... | It is better to work generating functions by multiplying by $z^n$ and sum over $n \ge 0$, which here gives:
$$
\frac{A(z) - a_0 - a_1 z}{z^2}
= 3 \frac{A(z) - a_0}{z}
- 2 A(z)
+ 2 \frac{1}{1 - z}
$$
This results in:
$$
A(z) = \frac{1 - 3 z + 4 z^2}{1 - 4 z + 5 z^2 - 2 z^3}
= \frac{2}{1 - 2 z} + \fra... | {
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"url": "https://math.stackexchange.com/questions/753074",
"timestamp": "2023-03-29T00:00:00",
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Finding $a$ and $b$ from $a^3+b^3$ and $a^2+b^2$ Question 1
Two numbers are such that the sum of their cubes is 14 and the sum of their squares is 6. Find the sum of the two numbers.
I did
$a^2+b^2=6$ and $a^3+b^3=14$ Find $a$ and $b$, two numbers. but got lost when trying to algebraicly solve it.
Thank you, Any help... | The problem doesn't specify any restriction on the numbers $a$ and $b$, so we're going to assume they are complex. Let $S=a+b$ denote the sum. Then
$$S^2=(a+b)^2=a^2+2ab+b^2=6+2ab$$
and
$$S^3=(a+b)^3=a^3+3a^2b+3ab^2+b^3=14+3abS$$
Rewriting the first of these as $2ab=S^2-6$ and multiplying both sides of the second by ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find value range of $2^x+2^y$ Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$
I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I can only find one solution pair of $4^x+4^y = 2^{x+1} + 2^{y+1}$. It seems very far from sol... | Let us denote $2^{x}=a,2^{y}=b,$where $a,b\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{+}$. Then we have $a^{2}+b^{2}=2(a+b)$ which is equivalent to $%
(a-1)^{2}+(b-1)^{2}=2.$So we take the part of the circle with center $M(1,1)
$ and radius $r=\sqrt{2}$ in the first district of the plane as t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/756213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 4
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Showing $\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64}$ I would like to show that
$$ \sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64} $$
I've been working o... | Use this formula (found here, and mentioned recently on MSE here):
$$\prod _{k=1}^{n-1}\,\sin \left({\frac {k\pi }{n}} \right)=\frac{n}{2^{n-1}} .$$
Let $n=13$, which gives $$\left(\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}}\right)\left(\sin{\frac{7\pi}{13}}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "7",
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Probability of number formed from dice rolls being multiple of 8
A fair 6-sided die is tossed 8 times. The sequence of 8 results is recorded to form an 8-digit number. For example if the tosses give {3, 5, 4, 2, 1, 1, 6, 5}, the resultant number is $35421165$. What is the probability that the number formed is a multip... | Here's a generalization:
The sample space for a six-sided die is $S = \{1,2,3,4,5,6\}$.
Theorem: There are exactly $\color{blue}{3}^n$ n-tuples $(x_{n-1}, x_{n-2}, \cdots, x_1, x_0)$ that satisfy $f(n) = 10^{n-1}x_{n-1} + 10^{n-2}x_{n-2} + \cdots + 10x_1 + x_0 \equiv 0 \mod{2^n}$ for $x_i \in S$
Proof by induction:
Ba... | {
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"timestamp": "2023-03-29T00:00:00",
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Solve $z^2 - iz = |z - i|$ I have the equation:
$z^2 - iz = |z - i|$
The solutions are $i$, $-\sqrt3/2 + i/2$, $\sqrt3/2 + i/2$
Can someone please walk me through or give me a hint...
| Here is a solution which is little different, algebraically, though it is "spiritually similar" to the work of Mario Gallegos:
given that
$z^2 - iz = \vert z - i \vert, \tag{1}$
we first take the modulus of each side, noting that
$\vert z^2 - iz \vert = \vert z(z -i) \vert = \vert z \vert \vert z -i \vert; \tag{2}$
w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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number of solutions to $x_1 + x_2 + x_3 + x_4 + x_5 = 31$ via generating function? I will be very happy to understand how to solve this problem with generating function:
How many solutions are there to the equation
$$x_1 + x_2 + x_3 + x_4 + x_5 = 31$$
where $x_i$ is a nonnegative integer,
and $x_2$ is even and $x_3$ i... | Non-negative $x_1$ corresponds to $\frac{z}{1 - z}$, $x_2$ even gives rise to $\frac{1}{1 - z^2}$, odd $x_3$ means $\frac{z}{1 - z^2}$, non-restriced $x_4$, $x_5$ are $\frac{1}{1 - z}$ each:
\begin{align}
[z^{31}] &\frac{z}{1 - z}
\cdot \frac{1}{1 - z^2}
\cdot \frac{z}{1 - z^2}
\cdot \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/759659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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One step Gauss Seidel method Apply one step of the Gauss Seidel method to $A\textbf{x} = b$ with
A = $\begin{bmatrix}
4 & 2 & 1 \\
1 & 4 & 1 \\
1 & 2 & 4
\end{bmatrix}$, b = $\begin{bmatrix}
4\\
5\\
8
\end{bmatrix}$ $x_{0}$ = $\begin{bmatrix}
64\\
64\\
-128
\end{bmatrix}$
Does the method converge to the s... | Yes, your first iteration is correct. Here are some iterates and details.
Iteration 1:
$$\left(
\begin{array}{cccccccccc}
1. \\ 33. \\ -14.75
\end{array}
\right)$$
Iteration 2:
$$\left(
\begin{array}{cccccccccc}
-11.8125 \\ 7.89063 \\ 1.00781
\end{array}
\right)$$
Iteration 3:
$$\left(
\begin{array}{cccccccccc}
... | {
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"timestamp": "2023-03-29T00:00:00",
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Solving $L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ priveded $a+b+c=0$
Let $a,b,c$ be such that $a+b+c=0$ and suppose that
$$L= \frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}.$$
Find the value of $L$.
I can only see the symmetry of these function but cannot solve it.
| Since $a+b+c|(a^3+b^3+c^3-3abc)$, $a+b+c = 0 \implies a^3+b^3+c^3 = 3abc$.
Then Wolfram Alpha says that this gives the sum as $1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find The Minimum Value of the quantity Find the minimum value of the quantity $$\frac{(a^2+3a+1)(b^2+3b+1)(c^2+3c+1)}{abc}$$,where $$a,b,c>0$$ and $$ a,b,c\in R $$are positive real numbers.
| Hint: $$\frac{(a^2+3a+1)(b^2+3b+1)(c^2+3c+1)}{abc} = \left(3+a+\frac1a \right)\left(3+b+\frac1b \right)\left(3+c+\frac1c \right)$$
Now can you show that $x + \dfrac1x$ has a minimum of $2$ for positive real $x$?
| {
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"timestamp": "2023-03-29T00:00:00",
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Number theory - rational number Are there any $x, y$ that fit in below
$\sqrt{4y^2-3x^2}$ such that an rational number is yielded.
Appreciate if explanation is given.
| We can find infinitely many solutions by considering the Pell equation $y^2-3s^2=1$.
We look for perfect squares $4y^2-3x^2$, where $x$ is even, say $2s$. Then we want $4y^2-12s^2$ to be a perfect square, say $(2t)^2$. So we want to solve the Diophantine equation $y^2-3s^2=t^2$.
There are already infinitely many solut... | {
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Is the modulus of i^n 1 for all n? As the tile says, is $\left | i^{n} \right | = 1$ for all real values of n?
| We note that $|i| = 1$: $|i| = |0+1i| = \sqrt{0^2+1^2} = \sqrt{1} = 1$.
Suppose that $|i^n| = 1$. Then, $|i^{n+1}| = |i^ni| = |i^n||i| = 1\cdot 1 = 1$. By induction, $|i^n| = 1$ for all positive integers $n$.
Another method:
In general, $|z|^2 = z\overline{z}$.
$$|i^n|^2 = (i^n)\overline{(i^n)} = \underbrace{i\ \overl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/767759",
"timestamp": "2023-03-29T00:00:00",
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How to find $\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}$? find the integral of $f(x)=\frac1{(x-1)^2\sqrt{x^2+6x}}$
my attempt =
$(x-1)=a$, $a=x+1$ so the integral'd be
$\int \frac {dx}{(x-1)^2\sqrt{x^2+6x}}=\int\frac{da}{a^2\sqrt{a^2+8a+7}} $
lets say $\sqrt{a^2+8a+7}=(a+1)t$
so $a=\frac{7-t}{t-1}$ and $da=\frac{-6dt}... | Put $$\frac{1}{x-1}=t$$ $\implies$ $$ \frac{dx}{(x-1)^2}=-dt$$ So
$$-I=\int \frac{t \,dt}{\sqrt{7t^2+8t+1}}$$ Put $$ 7t^2+8t+1=z^2$$ $\implies$
$$(7t+4)dt=zdz$$ So
$$ -7I=\int \frac{7t+4-4 \,dt}{\sqrt{7t^2+8t+1}}=\int \frac{7t+4 \,dt}{\sqrt{7t^2+8t+1}}-\int \frac{4 \,dt}{\sqrt{7t^2+8t+1}}$$$$$$ So
$$-7I=z-\int \frac{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/768020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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put numbered balls in four similar boxes of a specific capacity... With how many ways can we put $12$ numbered balls in $4$ similar(not numbered) boxes of capacity $3$ each one?
Is it maybe $3^4$ ?
| You can choose $3$ out of $12$ balls for the first box, $3$ out of $9$ for the second...and so on...But since the boxes are similar you should divide the result by all the possible combinations of the $4$ boxes ($4!$), so the number requested is:
$$\frac{\binom{12}{3}\cdot\binom{9}{3}\cdot\binom{6}{3}\cdot\binom{3}{3}... | {
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Is the following Markov Chain a martingale? Say I have a finite, ergodic Markov chain with states ${0,1,2,3}$ and with the following transition matrix:
$$\begin{bmatrix}
\frac{7}{10} & \frac{3}{10} & 0 &0\\
\frac{1}{10} & \frac{6}{10} & \frac{3}{10} &0\\
0 & \frac{3}{10} & \frac{6}{10} & \frac{1}{10} \\
0& 0& \f... | For every martingale $(M_n)$, $\mathbb{E}(M_{n+1} | M_n ) = M_n$. But here, $\mathbb{E}(M_{n+1}|M_n=0) = \frac{3}{50}$. Therefore, $(M_n)$ is not a martingale.
| {
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12 balls in a box to distribute by 2 bags. There is a box with 12 balls inside, 4 green, 5 blue and 3 black.
We want to take out 4 balls for each one of 2 bags (bag A and bag B), and we take note of the number of balls of each color that the bags have.
How many ways are there to distribute the balls?
| We could start by figuring out the number of ways that $4$ balls do NOT go in either bag (that would also determine the $8$ that do).
The possibilities are:
*
*4 green $\displaystyle \Rightarrow \dbinom{4}{4} = 1$
*3 green, 1 blue $\displaystyle \Rightarrow \dbinom{4}{3} \dbinom{5}{1} = 4\cdot5 = 20$
*3 green, 1 b... | {
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Why the linear numerator for fractions with irreducible denominators and constant numerators for reducible denominators? For example:
$\Large{\frac{2x^3+5x+1}{(x^2+4)(x^2+x+2)}}$ breaks down to $\Large{\frac{ax+b}{x^2+4}+\frac{cx+d}{x^2+x+2}}$
I have been told that since the denominators are irreducible, the numerators... | Because remainder of division is always smaller then the degree of the polynomial so we assume it's exactly 1 smaller,and find if the other terms are zero
$$\frac{2x^3+5x+1}{(x+2)(x-2)}=\frac{ax+b}{x+2}+\frac{cx+d}{x-2}=\frac{ax+b-a(x+2)}{x+2}+a+\frac{cx+d-c(x-2)}{x-2}+c=\frac{ax+b-ax-2a}{x+2}+\frac{cx+d-cx+2c}{x-2}+a+... | {
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Why does $\dfrac{2}{\sqrt{-4x^2-4x}}$ simplify into $\dfrac{1}{\sqrt{-x^2-x}}$?
Why does $\dfrac{2}{\sqrt{-4x^2-4x}}$ simplify into $\dfrac{1}{\sqrt{-x^2-x}}$?
What's going on here? How is it being simplified?
| $$\dfrac{2}{\sqrt{-4x^2-4x}}=\dfrac{2}{\sqrt{4(-x^2-x)}}=\dfrac{2}{\sqrt{4}\sqrt{(-x^2-x)}}\\
=\dfrac{2}{2\sqrt{(-x^2-x)}}=\dfrac{1}{\sqrt{(-x^2-x)}}$$
| {
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Finding an explicit formula for $a_n$ defined recursively by The sequence $a_n$ is defined recursively by $a_0=0$, $a_n=4a_{n-1}+1$. I must use generating functions to solve this. $n\geq1$.
I have found a pattern:
$$\sum_{n=1}^\infty(4a_{n-1}+1)x^n = x+5x^2+21x^3+85x^4+341x^5+\ldots$$
If we subtract 1 from each term, r... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
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Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares. I have figured out a (long, and tedious) way to do it. But I was wondering if there is some sort of direct correlation or another path that I completely missed.
My attempt at the program was as follows:
A number of the form, $8k +... | Note that $x^2\equiv (-x)^2\pmod 8$. So the squares mod $8$ are $0^2=0$, $1^2=1$, $2^2=4$ and $3^2=1$. It is evident that three of these numbers cannot add up $7$.
| {
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The proof of $e^x \leq x + e^{x^2}$ How can we prove the inequality $e^x \le x + e^{x^2}$ for $x\in\mathbb{R}$?
| $$e^{x} = 1 + x + \frac{x^2}{2!} + \dots \le 1 + x + 2 \left(\left( \frac{x}{2} \right )^2+\left( \frac{x}{2} \right )^3 +\dots \right)\le 1 + x+\left( \frac{x}{2} \right )^2 \left( \frac {4}{2 - x}\right)$$
$$x + e^{x^2} \ge x +1 + x^2 + \frac{x^4}{2!} = 1+ x+ \left( \frac{x}{2} \right )^2 2(2+x^2) \ge 1+ x+ 4 \l... | {
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Intersection multiplicity
Let $f=y^2-x^3$ and $g=y^3-x^7$. Calculate the intersection multiplicity of $f$ and $g$ at $(0,0)$.
I know the general technique for this (passing to the local ring) but I having difficulty with the fact that $3,7,2$ have no common factors.
| Using the properties of intersection number can save much time for the
calculation of intersection numbers.
If $P$ is a point, two affine plane curves $F$ and $G$ have no common
components, then: (see section 3.3 in Fulton's algebraic curves)
*
*$I(P,F\cap G)=I(F\cap (G+AF))$ for any $A\in k[X,Y]$;
*$I(P,F\cap G)=m... | {
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Find all real $x$ ,such $8x^3-20$ and $2x^5-2$ is perfect square
Find all real numbers $x$,such
$$8x^3-20,2x^5-2$$ is the perfect square of an integer
My idea: First we find the real number $x$ such $$8x^3-20,2x^5-2$$ is postive integer numbers,and second the real numbers such
$$8x^3-20=m^2,2x^5-2=n^5$$
How prov... | First observe that $x$ is a positive integer. $x^5$ and $x^3$ are rationals; then $x=(x^3)^2/x^5$ is rational; finally $(2x)^5$ is an even positive integer so $2x$ must be an even positive integer.
In order to find a bound for $x$, consider the product of $8x^3−20=m^2$ and $2x^5−2=n^2$ and complete the square.
\begin{... | {
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Polar Coordinates --- Equation of a line
Hey, does anyone know how to tackle this question? I've tried using the formula r=d*sec(theta-alpha)but I'm not sure what each of the variables are equal to. If anyone can offer any help, it would be greatly appreciated!
|
The most direct approach, which would be preferred were this an exam question, is to find the equation for the line in Cartesian coordinates in the manner one would in pre-calculus, and then transform that result into polar coordinates. The point $ \ (-11, -7) \ $ (which is on the circle of radius $ \ \sqrt{(-11)^2 +... | {
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Can someone explain the steps of manipulation of this equation for the value of x?
*
*$x = \sum_{j=1}^{n-1} 5j + 7$
*$x = \left(\sum_{j=0}^{n-1} 5j + 7 \right) - 7$
*$x = \left(5\sum_{j=0}^{n-1} j + \sum_{j=0}^{n-1}7 \right) - 7$
*$x = \left(5\sum_{j=0}^{n-1} j + 7n \right) - 7$
*$x = \frac{5n^2}{2} + 7n - 7$
I... | 4: What do you get if you do $7+7+...+7+7~~ n$ times? $~~7n!$
(from $0 $ to $ n-1$ there are n steps.)
For step 5 you have to write: $$x = \frac{5n^2-5n}{2} + 7n - 7$$
as$$\sum_{j=0}^{n-1} j=\sum_{j=1}^{n-1} j=\sum_{j=1}^{n} j-n=\frac{(n+1)n}{2}-n=\frac{n^2+n-2n}{2}=\frac{n^2-n}{2}$$
| {
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Count subsets whose cardinalities are congruent to 0, 1 and 2 modulo 3 respectively Given a set of N elements, compute the number of subsets whose cardinalities are congruent to 0, 1 and 2 modulo 3 respectively.
Any hints would be appreciated.
Thanks!
| You want to evaluate
\begin{align}a_N=\binom N0+\binom N3+\binom N6+\binom N9+\cdots,\\b_N=\binom N1+\binom N4+\binom N7+\binom N{10}+\cdots,\\c_N=\binom N2+\binom N5+\binom N8+\binom N{11}+\cdots.\end{align}
Note that, if $x^3=1$, then
\begin{align} (1+x)^N&=\binom N0+\binom N1x+\binom N2x^2+\binom N3x^3+\binom N4x... | {
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block matrix multiplication If $A,B$ are $2 \times 2$ matrices of real or complex numbers, then
$$AB = \left[
\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right]\cdot
\left[
\begin{array}{cc} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array} \right]
=
\left[
\begin{array}{cc} a_{11}b_{11}+a_{12}b_{21... | It is always suspect with a very late answer to a popular question, but I came here looking for what a compatible block partitioning is and did not find it:
For $\mathbf{AB}$ to work by blocks the important part is that the partition along the columns of $\mathbf A$ must match the partition along the rows of $\mathbf{B... | {
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Is my proof correct? (standard algebra, some calculus) I'm teaching myself mathematics and I tried to solve a problem from the STEP exam (Cambridge's admission test). The solution is considerably shorter and less "clumsy" than mine but I would like to know if my proof is at least correct.
Thank you.
We have $$f'(x)=a-... | Yes you have done it in the right sense .
| {
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$f(\frac1{1234}) + f(\frac3{1234}) + f(\frac5{1234}) + ..... + f(\frac{1231}{1234}) + f(\frac{1233}{1234}) = ?$ If $f(x) = \frac{e^{2x-1}}{1+e^{2x-1}}$, then how to evaluate $$f(\frac1{1234}) + f(\frac3{1234}) + f(\frac5{1234}) + ..... + f(\frac{1231}{1234}) + f(\frac{1233}{1234}) ?$$
I know I'm missing some trick here... | Using the trick you've identified,
\begin{align*}
f(x) + f(1 - x) &= \frac{e^{2x - 1}}{1 + e^{2x - 1}} + \frac{e^{2(1 - x) - 1}}{1 + e^{2(1 - x) - 1}} \\
&= \frac{e^{2x - 1}(1 + e^{1 - 2x}) + e^{1 - 2x}(1 + e^{2x - 1})}{(1 + e^{2x - 1})(1 + e^{1 - 2x})} \\
&= \frac{e^{2x - 1} + 1 + e^{1 - 2x} + 1}{1 + e^{2x - 1} + e^{1... | {
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Is there an elegant way to simplify $\frac{\tan(x+20^{\circ })-\sin(x+20^{\circ })}{\tan(x+20^{\circ })+\sin(x+20^{\circ })}$ I wonder how to solve this equation: $$\frac{\tan(x+20^{\circ })-\sin(x+20^{\circ })}{\tan(x+20^{\circ })+\sin(x+20^{\circ })}=4\sin^{2}\left(\frac{x}{2}+10^{\circ }\right)$$ in an elegant/short... | $4sin^2(\frac{x + 20^\circ}2) = 4\cdot1/2\cdot(1-\cos(x+20^\circ)) = 2-2\cos(x+20^\circ)$
Applying componendo and dividendo on $\frac{tg(x+20^{\circ })-sin(x+20^{\circ })}{tg(x+20^{\circ })+sin(x+20^{\circ })}=4sin^{2}(\frac{x}{2}+10^{\circ })$ ,
$-\sec(x+20^\circ) = \frac{2-2\cos(x+20^\circ)+1}{2-2\cos(x+20^\ci... | {
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Simplifying $\sum_{k=1}^n (n-k)2^{k-1}$ via combinatorics in Aigner, A course in Combinatorics I must simplify $\sum_{k=1}^n (n-k)2^{k-1}$ using the rule of sum. How might I go about doing this?
Thank you and regards.
| $\sum_{k=1}^n(n-k)2^{k-1} = \sum_{k = 1}^n n2^{k-1} - \sum_{k = 1}^n k2^{k-1}$.
In the first sum, we can factor out $n$ from each term so $\sum_{k=1}^n n2^{k-1} = n(1+2+ \cdots + 2^{n-1}) = n(2^n-1)$.
For the second term, we use a standard calculus argument. $1+x+x^2+ \cdots + x^k = \frac{x^{k+1}-1}{x-1}$.
Then differe... | {
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A mathematical phenomenon regarding perfect squares.... I was working on identifying perfect squares for one of my programs regarding Pythagorean triplet.
And I found that for every perfect square if we add its digits recursively until we get a single digit number, e.g. 256 -> 13 -> 4 etc. we get the single digit as e... | The standard observation is that "add the digits of a number recursively" is essentially just a fancy way to say "find the value of the number modulo 9".
So your question is really asking "which numbers are squares modulo 9?" This is easy to answer with modular arithmetic: since there are only 9 numbers modulo 9, we ca... | {
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Number theory problems. It is given: $3m+1=$perfect square. Express $m+1$, as the sum of $3$ perfect squares.
I tried to solve the problem by checking for odd and even values perfect squares $4k^2,{(2k+1)}^2$. I got somewhat convincing form of result from $3m+1={(2k+1)}^2$:
$m+1=\dfrac43(k^2)+\dfrac43(k)+1,$ so that ... | $3m + 1 = n^2$. This means: $n = 3k - 1$ or $n = 3k + 1$. If $n = 3k + 1$ then:
$m = \dfrac{n^2 - 1}{3}$, so $m + 1 = \dfrac{n^2 + 2}{3} = \dfrac{(3k + 1)^2 + 2}{3} = \dfrac{9k^2 + 6k + 3}{3} = 3k^2 + 2k + 1 = k^2 + k^2 + (k+1)^2$.
The other case is similar.
| {
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Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$ $f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies
$$
f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0.
$$
Can someone give explicit examples of $f$, apart from... | Here is the set of all possible $f$.
The algebra gets a little gross, so bear with me.
First, let
$f_1: \left[1 - \frac{\sqrt{3}}{2}, 1 \right]
\to \mathbb{R}$ be an arbitrary continuous function such that $f_1(x) = 0$ on the closed interval $\left[ \frac{\sqrt{3}}{2}, \frac{7}{4} - \frac{\sqrt{3}}{2} \right]$,
and $f... | {
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Find all values of the parameter $a$ so that $f(x)= y$ is always positive Find all values of the parameter $a$ so that $y>0$.
The equation is:
$$y = (a − 1)x^2 − (a + 1)x + a + 1$$
Thanks!
| First of all, the discriminant should be negative but $a-1>0$.
We get
$$(a+1)^2 - 4(a-1)(a+1) < 0 \Longleftrightarrow a^2+2a+1 - (4a^2 - 4) = -3a^2+2a +5 < 0.$$
By using quadratic formula
$$-3a^2+2a+5 = (-3a +5 )(a + 1) < 0.$$
The formula above will have negative value when $a < -1$ or $a > \frac{5}{3}$.
Because the va... | {
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Floor function inequality: $\frac{n^2}{x}-\left\lfloor\frac{n^2}{x}\right\rfloor+\frac{2n+1}{x}-\left\lfloor\frac{2n+1}{x}\right\rfloor<1$ I would like to dissect the following inequality to figure out its properties.
$$\frac{n^2}{x}-\left\lfloor\frac{n^2}{x}\right\rfloor+\frac{2n+1}{x}-\left\lfloor\frac{2n+1}{x}\right... | (Getting this off the unanswered list.)
For any $r\in\Bbb R$, $r-\lfloor r\rfloor$ is the fractional part of $r$, usually written $\{r\}$. Thus, you’re asking about
$$\left\{\frac{n^2}x\right\}-\left\{\frac{2n+1}x\right\}\,.$$
Clearly
$$0\le\left\{\frac{n^2}x\right\},\left\{\frac{2n+1}x\right\}\le\frac{x-1}x=1-\frac1x<... | {
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Evaluation of $\int_0^1 \frac{\log^2(1+x)}{x} \ dx$ One of the ways to approach it lies in the area of the dilogarithm, but is it possible to evaluate it
by other means of the real analysis (without using dilogarithm)?
$$\int_0^1 \frac{\log^2(1+x)}{x} \ dx$$
EDIT: maybe you're aware of some easy way to do that. I'd a... | On the path of Felix Marin,
\begin{align}J&=\int_0^1 \frac{\ln(1+x)^2}{x}\\
&\overset{y=\frac{1}{1+x}}=\int_{\frac{1}{2}}^1 \frac{\ln^2 x}{x(1-x)}\,dx\\
&=\int_{\frac{1}{2}}^1 \frac{\ln^2 x}{x}\,dx+\int_{\frac{1}{2}}^1 \frac{\ln^2 x}{1-x}\,dx\\
&=\frac{1}{3}\left(\ln^3 (1)-\ln^3\left(\frac{1}{2}\right)\right)+\int_0^1 ... | {
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Find the minimum value of $P=\frac{1}{4(x-y)^2}+\frac{1}{(x+z)^2}+\frac{1}{(y+z)^2}$ Let $x,y,z$ be real numbers such that $x>y>0, z>0$ and $xy+(x+y)z+z^2=1$.
Find the minimum value of $$P=\frac{1}{4(x-y)^2}+\frac{1}{(x+z)^2}+\frac{1}{(y+z)^2}$$
I tried using some ways, but failed. Please give me an idea. Thank you.
| From @DeepSea's above, suppose $b\equiv f(\,a\,),\,ab= 1\,\therefore\,{b}'= -\,\dfrac{b}{a},\,(\,a- 2\,b\,)(\,a- 2^{\,(\,\frac{1}{2}\,)}\,)\geqq 0$.
$$a- b> 0 \tag{assume}$$
Let $$W(\,a\,)= \frac{1}{4(\,a- b\,)^{\,2}}+ \frac{1}{a^{\,2}}+ \frac{1}{b^{\,2}}- 3$$
$$\therefore\,{W}'(\,a\,)= -\,\frac{2}{a^{\,3}}+ \frac{{b}... | {
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Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$.
Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$.
I have computed $$c_n = \frac 1 {2\pi} \int^{\pi}_{-\pi} |\sin(x)|e^{-inx} dx = \frac 1 {... | The exponential Fourier series for this particular $f(x)$ is
\begin{align}
|\sin(ax)| = \sum_{n= -\infty}^{\infty} c_{n} e^{-i nx},
\end{align}
where
\begin{align}
c_{n} = \frac{1}{2\pi} \int_{-\pi}^{\pi} |\sin(ax)| \ e^{-i nx} dx.
\end{align}
This integral can be calculated as follows.
\begin{align}
c_{n} &= \frac{1}... | {
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Integral $I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3}$ Hi how can we prove this integral below?
$$
I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3}
$$
I tried to use
$$
I=\int_0^1 \frac{\log^2x}{1-x(1-x)}\mathrm dx
$$
and now tried changing variables to $y=x(1-x)$ ... | Here is another way to do it. Let $$I=\int_{0}^{1} \frac{(\ln(x))^2}{x^2-x+1} \ dx =\int_{0}^{1} \frac{(\ln(x))^2 (1+x)}{1+x^3} \ dx.$$ By a change of variables $x=\frac{1}{u}, \ dx =-\frac{1}{u^2} \ du,$ we have
$$I=\int_{1}^{\infty} \frac{(\ln(u))^2 (1+u)}{1+u^3} \ du,$$ which implies
$$I=\frac{1}{2} \int_{0}^{\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 8,
"answer_id": 7
} |
Inequality $\sum\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum \frac{1}{x + n^2} $
$x\geq0$, then, we have
$$\sum_{n=1}^{\infty}\frac{x}{(x + n^2)^2}<\frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{x + n^2} $$
The problem is not easy, even $x=1$. Any help will be appreciated
| I think this approach is also valid, and simpler.
A term-wise subtraction (LHS - RHS) gives:
$$
\frac{x}{(x+n^2)^2} - \frac{1}{2(x+n^2)} = \frac{2x - (x+n^2)}{2(x+n^2)^2} = \frac{x-n^2}{2(x+n^2)^2}
$$
for sufficiently large number of terms, this can be proved to be less than zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/803497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Trigonometry substitution Given the expansion of $\cos(5x) = 16\cos^5(x)-20\cos^3(x)+5\cos(x)$. How can I find the find the value of $\cos(\frac{\pi}{10})$ and $\cos(\frac{7\pi}{10})$ using $\cos(5x) = 0$ ? I know that I can change $\cos (5x)$ as a quartic equation and can get $\cos(x) = \pm\sqrt \frac{5 \pm\sqrt 5}{8}... | First of all, $$0<\frac\pi{10}<\frac\pi2<\frac{7\pi}{10}$$
So, $\displaystyle\cos\frac\pi{10}>0;\cos\frac{7\pi}{10}<0$
$\displaystyle\implies-\cos\frac{7\pi}{10}=\cos\left(\pi-\frac{7\pi}{10}\right)=\cos\frac{3\pi}{10}>0$
and using Prosthaphaeresis Formulas, $\displaystyle\cos\frac\pi{10}-\cos\frac{3\pi}{10}=2\sin\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How can I find the following product? $ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$ How can I find the following product using elementary trigonometry?
$$ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$$
I have tried using a substitution, but nothing has worked.
| tan 20 tan 40 tan 80 = (sin 20 sin 40 sin 80) / (cos 20 cos 40 cos 80)
Solving for numerator :
sin 20 sin 40 sin 80 = (sin 80 sin 20)(sin 60 sin 40) / (sin 60)
= (1/2)(cos 60 - cos 100)(1/2)(cos 20 - cos 100) / (sin 60) ... (product rule)
= (1/4)(1/2 + sin 10)(cos 20 + sin10) / (sin 60) ... (cos 60 = 1/2 & cos 10... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prove that $A^k = 0 $ iff $A^2 = 0$ Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$.
I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this question comes before this.
Thank you very much for your help!
| A^k=0 iff A^2=0 k positive interger >2 you can write k=2+n and n>=1. Using induction: A^2+n=0 Then A^2=0. Take A^3=A^2A=0. And prove A^2+n+1=0 =>A^2=0 and A^2+nA=0. Then A^2=0.
First direction A^2=0 then A^k=0.
You can take A and B as a matrix.. Lets represent both matrix . We have A=[a b over c d] and B=[b1 b2 over b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 8,
"answer_id": 7
} |
Linear Algebra determinant reduction Prove, without expanding, that
\begin{vmatrix}
1 &a &a^2-bc \\
1 &b &b^2-ca \\
1 &c &c^2-ab
\end{vmatrix} vanishes.
Any hints ?
| Simply add a multiple of the first column to the last
$$\begin{vmatrix} 1 & a & a^2-bc\\1&b&b^2-ac\\1&c&c^2-ab\end{vmatrix}
=\begin{vmatrix} 1 & a & a^2-bc+(1)(ab+ac+bc)\\1&b&b^2-ac+(1)(ab+ac+bc)\\1&c&c^2-ab+(1)(ab+ac+bc)\end{vmatrix}
= \begin{vmatrix} 1 & a & a(a+b+c)\\1&b&b(a+b+c)\\1&c&c(a+b+c)\end{vmatrix}
= 0$$
The... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Show that the equation $2a^4+2a^2b^2+b^4=c^2$ hasn't an integer solution Question:
Prove that: if $a,b,c\in Z$,and $a\neq 0$ show that: the equation
$$2a^4+2a^2b^2+b^4=c^2$$ has no solution.
My idea:
since
$$a^4+(a^2+b^2)^2=c^2$$
and I want use Pythagorean triple
and let $$a^2=m^2-n^2. a^2+b^2=2mn, c=m^2+n^2,(m,n)... | Let $(a,b,c)$ be a solution of
$$ 2a^4 + 2a^2b^2 + b^4 = c^2.$$
By evaluating the different values of $a,b,c$ modulo $4$ (Edit: $8$), one can easily show that $a$ must be even. If $b$ would also be even, then $(\frac a2, \frac b2, \frac c4)$ is also a solution to the equation. Hence if $a\neq 0$ we may assume that $b$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 1,
"answer_id": 0
} |
How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$?
How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$ ?
The numerator is a irreducible polynomial so I can't use partial fractions.
I tried the substitutions $t = x^2, t=x^4$ and for the formula $\int u\,dv = uv - \int v\,du$ I tried using: $u=\frac{x^4 + 1 }{x^6 + 1} , \,dv=\,d... | Hint: Decompose the denominator using $a^3+b^3=(a+b)(a^2-ab+b^2)$. Then add $\&$ subtract $x^2$ in the numerator.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/811911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Compute $\int_0^{\pi/4}\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)} x\exp(\frac{x^2-1}{x^2+1}) dx$
Compute the following integral
\begin{equation}
\int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, \exp\left[\frac{x^2-1}{x^2+1}\right]... | From the numerator, collect the logarithmic terms first.
$$\displaystyle \int_0^{\pi/4} x\frac{(1+x^2)+(x^2-1)\ln\left(\frac{1-x^2}{1+x^2}\right)}{(1-x^4)(1+x^2)}\exp\left(\frac{x^2-1}{x^2+1}\right)\,dx$$
Rewrite $(1-x^4)=(1-x^2)(1+x^2)$ and divide the numerator by $(1+x^2)$.
$$\displaystyle \int_0^{\pi/4} \frac{x}{(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/815863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "116",
"answer_count": 2,
"answer_id": 0
} |
Quick method for finding eigenvalues and eigenvectors in a symmetric $5 \times 5$ matrix? The matrix $B$:
$B =
\pmatrix{
0 & 0 & 0 & 0 & 0 \cr
0 & 8 & 0 & -8 & 0 \cr
0 & 0 & 8 & 0 & -8 \cr
0 & -8 & 0 & 8 & 0 \cr
0 & 0 & -8 & 0 & 8 \cr
}$
Which has nonzero eigenvalues $\lambda_1=16$ and $\lambda_2=... | Hint: if $A$ and $B$ are square of the same order, $$\det\begin{pmatrix} A & B \\ B & A\end{pmatrix}=\det(A-B)\det(A+B)$$
So setting
$$A=\begin{pmatrix}8-\lambda & 0 \\ 0 & 8-\lambda\end{pmatrix}$$
and
$$B=\begin{pmatrix}-8 & 0\\0&-8\end{pmatrix}$$
shows that the determinant of your $4\times 4$ matrix is $(16-\lambd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/816087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 1
} |
Can One Integrate $\frac{1}{i} \int \frac{e^{ix}-e^{-ix}}{e^{ax}+e^{-ax}+e^{ix}+e^{-ix}}dx$ I'd like to integrate
$$I(a)=\int \frac{\sin(x)}{\cosh(ax)+\cos(x)}dx$$
by changing $sin$ etc... into their exponential representation.
Using $e^{ix} = \cos(x) + i \sin(x)$ and $e^{ax} = \cosh(ax)+\sinh(ax)$ we have
$$\frac{1}... | Consider the integral
\begin{align}
I(a) = \int \frac{\sin(x) }{ \cos(x) + \cos(ax) } \ dx.
\end{align}
This integral can be readily evaluated as is given by
\begin{align}
I(a) = \frac{1}{a^{2}-1} \ \left[ (a+1) \ln\left( \cos\left(\frac{(1-a)x}{2}\right)\right) - (a-1) \ln\left( \cos\left(\frac{(1+a)x}{2}\right)\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/817519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Simple algebra formula for which I can't find the right answer I have the formula $y + (z + 1) = \frac{1}{2} \cdot (z + 1) \cdot (z + 2)$, and I should work to $y = \frac{1}{2}\cdot z \cdot (z + 1)$.
Somebody showed me how it's done:
$y + (z + 1) = \frac{1}{2} \cdot (z + 1) \cdot (z + 2)$
$y + (z + 1) = \frac{1}{2} \cd... | There is an error in the fourth line of your second set of equations:-
$$y = \frac{1}{2} \cdot z^2 + 3z + 2 - z - 1$$
should be
$$y = \frac{1}{2} \cdot (z^2 + 3z + 2 \color{red}{- 2z - 2})\\\Rightarrow y = \frac{1}{2} \cdot (z^2 + z)=\frac{1}{2}\cdot z(z+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/818475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Linear algebra, power of matrices $P^{-1}AP =
\begin{pmatrix}
-1 & 1 & 0 & 0 \\
0 & -1 & 1 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 2 \\
\end{pmatrix}
$
with
$P=
\begin{pmatrix}
-1 & 1 & 0 & 0 \\
0 & -1 & 1 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
$
and $P^{-1}$ is the inverse of $P$
Find $... | Because of the way that $P$ and $A$ are constructed, it is not hard to see that $PA=AP$. Consequently, $P^{-1}AP=P^{-1}PA=A$. So that makes life easier. You can then write
$$
A =
\left[\begin{array}{cccc}
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/819222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
$\sum_{k=1}^{\infty} \frac{a_1 a_2 \cdots a_{k-1}}{(x+a_1) \cdots (x+a_k)}$ Hey guys I was reading Alfred van der Poortens paper regarding Apery's constant and I came across this pretty equality. For all $a_1, a_2, \ldots$
\begin{align}
\large\sum_{k=1}^{\infty}\frac{a_1a_2\cdots a_{k-1}}{(x+a_1)\cdots (x+a_k)} = \frac... | $$(x+a_1)(x+a_2)(x+a_3)(x+a_4)(x+a_5)... =A(x)\tag 1$$
$$\frac{A(x)}{A(x)}=1=x\frac{(x+a_2)(x+a_3)(x+a_4)(x+a_5)...}{(x+a_1)(x+a_2)(x+a_3)(x+a_4)(x+a_5)... }+a_1\frac{(x+a_2)(x+a_3)(x+a_4)(x+a_5)...}{(x+a_1)(x+a_2)(x+a_3)(x+a_4)(x+a_5)... } \tag 2$$
$$1=\frac{x}{(x+a_1)}+a_1\frac{(x+a_2)(x+a_3)(x+a_4)(x+a_5)...}{(x+a_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Stuck on finding the value of $\sum_{n=0}^\infty {n(n+1) \over 3^n}$ I am trying to find the value of the series
$$
\sum_{n=0}^\infty {n(n+1) \over 3^n}
$$
Here's what I have done so far:
$$
\sum_{n=0}^\infty {n(n+1) \over 3^n}=\sum_{n=0}^\infty {n^2 \over 3^n}+\sum_{n=0}^\infty {n \over 3^n}
$$
Determining the values... | $|x|<1:\\\sum_{n=0}^{\infty }x^{n}=\frac{1}{1-x}\Rightarrow 1+\sum_{n=1}^{\infty }x^{n}=\frac{1}{1-x}\\\Rightarrow \sum_{n=1}^{\infty }x^{n}=\frac{x}{1-x}\\\frac{d}{dx}\sum_{n=0}^{\infty }x^{n}=\frac{d}{dx}\frac{1}{1-x}\Rightarrow \sum_{n=1}^{\infty }nx^{n-1}=\frac{1}{(1-x)^2}\Rightarrow \sum_{n=1}^{\infty }nx^{n}=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$? Consider the real $n \times n$-matrices $A$ and $B$.
Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$ ?
| Yes, $A$ and $B$ can fail to commute. Consider
$$
A = \left[\begin{array}{@{}rrr@{}}
0 & 1 & 0 \\
-1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right],\qquad
B = \left[\begin{array}{@{}ccc@{}}
0 & -\tfrac{1}{2} & \tfrac{\sqrt{3}}{2} \\
\tfrac{1}{2} & 0 & 0 \\
-\tfrac{\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/825934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Two ways to show that $\sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ Show that: $\large \sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ on: $0<x<\frac {\pi}2$
I tried to solve it in two ways and got a little stuck:
One way is to use Cauchy's MVT, define $f,g$ such that $f(x)=\sin x -x +\frac {x^3}{3!}$ and $g(x)=\frac ... | Using the Mean value theorem is the right idea. You need the MVT to prove the following useful lemma:
If $f'(x)>g'(x)$ for all $a\ge x>0$ and $f(0)=g(0)$, then $f(x)>g(x)$ for all $a\ge x>0$.
Letting $g(x)=\sin(x)$ and $f(x)=x-\frac{x^3}{3!} +\frac{x^5}{5!}$ and then iterating this lemma will give you your answer. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/826015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Writing $2$ roots of a cubic in terms of the third root Let $\theta$ be a root of $x^3-3x+1$. Since the discriminant is a square, the splitting field of this polynomial is just $\mathbb Q(\theta)$. Now, I want to write the other roots as linear combinations of $1$, $\theta$, and $\theta^2$.. If we let $\alpha$ and $\be... | $(a+b\theta+c\theta^2)^2 = 12-3\theta^2$ and $\theta^3 = 3\theta-1$ (from the initial polynomial, and the fact that plugging in $\theta$ will give zero).
Multiplying out gives
$\begin{align*}a^2+2ab\theta+(2ac+b^2)\theta^2+2bc\theta^3+c^2\theta^4 & = a^2+2ab\theta+(2ac+b^2)\theta^2+2bc(3\theta-1)+c^2(3\theta-1)\theta \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/832442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Help with Evaluating a Logarithm A precalculus text asks us to evaluate $\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{256}}{\sqrt[6]{32}}$
I do the following:
$\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{(2^2)^3\cdot 2^2}}{\sqrt[6]{2^3\cdot 2^2}}$
$\equiv \log_{8}\dfrac{\sqrt{2}\cdot 2^2\cdot\sqrt[3]{2^2}}{\sqrt{2}\cdot\sqrt[6]{2^2}}... | $$ \log_{8}\left(\frac{2^2\cdot\sqrt[3]{2^2}}{\sqrt[3]{2}}\right) = \log_8 \left(2^2\cdot \sqrt[3]2\right) = \log_8 2^{7/3} = \frac 73 \log_8(2)= \frac 73 \cdot \frac 13 = \frac 79$$
$$8^{1/3} = 2 \iff \log_8 2 = \frac 13$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/833200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Help Evaluating $\lim_{x \to \infty} \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}$ Does anyone know how to evaluate the following limit?
$$
\lim_{x \to \infty} \sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}
$$
The answer is $\frac{1}{2}$, but I want to see a step by step solution if possible.
| $$\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}=\frac{\sqrt{x+\sqrt{x}}}{\sqrt{x+\sqrt{x+\sqrt{x}}}+\sqrt{x}}=\frac{\sqrt{1+\frac{1}{\sqrt{x}}}}{\sqrt{1+\sqrt{\frac{1}{x}+\frac{1}{x^{\frac{3}{2}}}}}+1}$$
I got the above by first multiplying by the conjugate to $\sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x}$ then dividing both the numerato... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/834471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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How does Wolfram get from the first form to the second alternate form? So, I was trying to compute an integral but I couldn't actually manage getting anywhere with it in its initial form. So, I inserted the function in Wolfram Alpha and I really got a nicer form (second alternate form). But I want to understand how tha... | $\dfrac{1}{(e^t + 1)(e^t + 2)(e^t + 3)} = \dfrac{1}{(x + 1)(x + 2)(x + 3)}$, where $x = e^t$.
Let $\dfrac{1}{(x + 1)(x + 2)(x + 3)} = \dfrac{A}{x + 1} + \dfrac{B}{x + 2} + \dfrac{C}{x + 3}$.
Then $A(x + 2)(x + 3) + B(x + 1)(x + 3) + C(x + 1)(x + 2) = 1$.
For $x = -1$:
$A(-1 + 2)(-1 + 3) = 1 \Rightarrow 2A = 1 \Rightarr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/834762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$? $f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and:
$$f(\frac1{x+1})\cdot x=f(x)-1$$
Does there exist such a function, if they do, are there infinitely many? And is there any explicit example? I have been able to derive th... | Such a function doesn't exist.
Forget $f(1) = 1$. Just assume $f(x)$ satisfy the functional equation on $[0,\infty)$. We have
$$
\begin{align}
f(x) &= 1 + x f\left(\frac{1}{x+1}\right)\\
&= 1 + x \left[ 1 + \frac{1}{x+1}f\left(\frac{x+1}{x+2}\right)\right]\\
&= 1 + x \left[ 1 + \frac{1}{x+1} + \frac{1}{x+2} f\left(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/836175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 0
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Help in Solving a Trigonometric Equation Solve the equation $$\left(\sin x + \cos x\right)^{1+\sin(2x)} = 2$$ when $-\pi \le x \le \pi $ .
I have tried to use $\sin (2x) = 2\sin x \cos x$ identity but I this doesn't lead me to a conclusion.
I will appreciate the help.
| To solve $(\sin x+\cos x)^{1+\sin 2x}=2$ when $−\pi\leq x\leq \pi$.
Let $z=\sin x+\cos x$, then $z^2=1+\sin 2x$. Hence the equation is equivalent to $z^{z^2}=2$ which has the only possible solution $z=\sqrt{2}$.
Hence $\sin x+\cos x=\sqrt{2}\sin(x+\frac{\pi}{4})=\sqrt{2}$.
Thus $\sin(x+\frac{\pi}{4})=1$, which implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Range of the function $f(x) = \frac{x^2+14x+9}{x^2+2x+3}\;,$ where $x\in \mathbb{R}$ Calculation of Range of the function $\displaystyle f(x) = \frac{x^2+14x+9}{x^2+2x+3}\;,$ where $x\in \mathbb{R}$
(Can we solve it Using $\bf{A.M\geq G.M}$) Inequality.
$\bf{My\; Try::}$ Let $\displaystyle y = f(x) = \frac{x^2+14x+9}{x... | Hint:
try to find a line $y=k$ such that
$$
\frac{x^2+14x+9}{x^2+2x+3} - k = 0
$$
has only one solution, that $k$ is a max (above the max there is no solution, below there are two solutions) or a min (below the min there is no solution, abore there are two solutions).
Thus
$$
\frac{[1-k]x^2+2[7-k]x+[9-3k]}{x^2+2x+3} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/845952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Problem of quadratic equation If $\alpha$ be a root of $ 4x^2 +2x -1 = 0 $ , prove that the other root is $4\alpha^3 - 3\alpha$ . I have tried to do it but of no success.[$4\alpha^3 -2\alpha$ = $\dfrac {-1}{2}$ and $4\alpha^4 - 3\alpha^2$ = $\dfrac {-1}{4}$ ] .How to prove it?
| If $\alpha$ is a root then
$$
4 \alpha^2 + 2 \alpha - 1 = 0.
$$
Therefore
$$
\begin{eqnarray}
4 \Big( -\tfrac{1}{2} - \alpha \Big)^2 + 2 \Big( -\tfrac{1}{2} - \alpha \Big) - 1 &=&
4 \Big( \tfrac{1}{4} + \alpha + \alpha^2 \Big)
- 2 \Big( \tfrac{1}{2} + \alpha \Big) - 1\\
&=& 4 \alpha^2 + 2 \alpha - 1\\
&=& 0.
\end{eqna... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/846203",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Algebraic inequality:$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac1a+\frac1b+ \frac1c$ Prove that :
If a,b,c $\in \mathbb{R^+}$
$$\frac a{b^2} + \frac b{c^2} +\frac c{a^2} \geq \frac1a+\frac1b+ \frac1c$$
My attempt :
We know that the sequence {a,b,c} and {$\frac1{a^2},\frac1{b^2},\frac1{c^2}$} are oppositely ... | $\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\left(\dfrac{a}{b^{2}}+\dfrac{b}{c^{2}}+\dfrac{c}{a^{2}}\right)\geq\left(\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{a}\right)^{2}$ : Cauchy-Schwarz
$\therefore$ $\dfrac{a}{b^{2}}+\dfrac{b}{c^{2}}+\dfrac{c}{a^{2}}\geq \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/846696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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using residue for integration Hi how do u calculate the integral which have square root ? for example for this integral (because of branches points I always baffle) :
$$\int_0^1 \frac{(1-x)^{1/4}\, x^{3/4}}{5-x}\, dx$$
| Consider $$f(z) = \frac{z^{3/4}(z-1)^{1/4}}{5-z} = \frac{|z|^{3/4}e^{3i/4 \arg(z)} |z-1|^{1/4} e^{i/4 \arg(z-1)} }{5-z}$$
where $ 0 \le \arg(z), \arg(z-1) < 2 \pi$.
By omitting the line segment $[0,1]$, $f(z)$ is well-defined on the complex plane and real-valued on the real axis for $x > 1$.
Then integrating clockwise ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/847317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What's the integral of $\int_0^\infty \frac{dx}{(x^4+1)^5}$ $$\int_0^\infty \frac{dx}{(x^4+1)^5}$$
My answer would be : $\dfrac{\Gamma(\tfrac{1}{4})\Gamma(\tfrac{19}{4})}{4\Gamma(5)}$
Solution: You can use this technique. – Mhenni Benghorbal
| Let $x = \sqrt{\tan \theta}$. Then, $dx = \dfrac{1}{2\sqrt{\tan \theta}}\cdot \sec^2 \theta d\theta =\dfrac{d\theta}{2\sin^{1/2}\theta\cos^{3/2} \theta}$. If $x = 0$, $\theta = 0$ and if $x \to \infty \ \Rightarrow \ \theta = \pi/2$.
$I =\int_{0}^{\infty}\dfrac{dx}{(x^4 + 1)^5} = \dfrac{1}{2}\int_{0}^{\pi/2}\dfrac{\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/847905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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what is wrong in my answer and what will be the correct solution for this probability question ? A and B play a game where each is asked to select a number from 1 to 5. If
the two numbers match, both of them win a prize. The probability that they
will not win a prize in a single trial is -
a)1/25 b) 24/25 c)2/25 d)2... | The rationale behind the solution is this:
The game conductor has a particular number from 1 to 5 that he wants A and B to pick. Let us say "2". Then
P(A picking 2) $= \frac{1}{5}$
P(B picking 2) $= \frac{1}{5}$
P(A picking other than 2) $= \frac{4}{5}$
P(B picking other than 2) $= \frac{4}{5}$
P(Both picking 2) $= \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/849233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find A,B,C,D consecutive numbers based on written addition formula $A, B, C, D$ are consecutive digits: $B$ is greater by $1$ than $A$, $C$ is greater by $1$ than $B$, $D$ is greater by one than $D$.
Four $X$s are digits $A,B,C,D$ in unknown order. Find $A,B,C,D$
\begin{matrix}
& A & B & C & D\\
& D & C & B & A\... | $ABCD+DCBA = 1111\cdot(2A+3)$
If $A=1$, $12300-5\cdot1111=6745$.
If $A=2$, $12300-7\cdot1111=4523$.
If $A=3$, $12300-9\cdot1111=2301$.
The next three possibilities will be too small, as we need a four digit number. So $A=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/849307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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numeric solutions on quadric surfaces Maybe it's a trivial thing, but I can't seem to find solution I'm looking for. I need to find a parametric solution to the following equation ($\mathbf{A}$ is positive definite):
$$
\mathbf{x}^T\mathbf{A}\mathbf{x}+\mathbf{b}^T\mathbf{x}+c=0
$$
How to find $x$, then, in some parame... | Well you can factor the problem by completing the square as follows,
\begin{align}
x^T A x + b^T x + c &= (x^T A x + b^T x + \frac{b^TA^{-1}b}{4}) + c - \frac{b^TA^{-1}b}{4} \\
&= (x^T A^{1/2} + \frac{1}{2}b^T A^{-1/2})(A^{1/2} x + \frac{1}{2}A^{-1/2} b) + c - \frac{b^TA^{-1}b}{4} \\
&= (x + \frac{1}{2}A^{-1} b)^T A (x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/849925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximizing Area of Triangle in Circle I was playing around with another example that I made up where I am trying to maximize the area of a triangle inscribed in a circle of radius. I want to do the problem using the method of Lagrange Multipliers.
Attempt: Consider the circle of radius 2 given by $x^2+y^2=4$. The fun... | The area is wrong. For a triangle inscribed in a circle (WLOG), you can assume one axis is parallel to the $x$ axis, say at height $y$. Then, the three points are:
$$\left(\sqrt{4-y^2},y\right),\ \left(-\sqrt{4-y^2},y\right), (x_3, y_3)$$
The area is therefore: $$A=(y_3-y)\sqrt{4-y^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/851809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
If $ x^2+y^2+z^2 =1$ for $x,y,z \in \mathbb{R}$, then find maximum value of $ x^3+y^3+z^3-3xyz $. If $ x^2+y^2+z^2 =1$, for $x,y,z \in \mathbb{R}$, what is the maximum of
$ x^3+y^3+z^3-3xyz $ ?
I factorize it... Then put the maximum values of $x+y+z$ and min value of $xy+yz+zx$...
But it is wrong as they don't hold si... | because we find the maximum,so let $A>0$
let $$x^3+y^3+z^3-3xyz=A>0\Longrightarrow (x+y+z)(x^2+y^2+z^2-xy-yz-xz)=A$$
so
$$x^2+y^2+z^2=\dfrac{A}{x+y+z}+xy+yz+xz=\dfrac{A}{x+y+z}+\dfrac{1}{2}[(x+y+z)^2-x^2-y^2-z^2]$$
By AM-GM inequality
\begin{align*}\Longrightarrow \dfrac{3}{2}(x^2+y^2+z^2)&=\dfrac{A}{x+y+z}+\dfrac{1}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/852186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
} |
A series converging (or not) to $\ln 2$ I have come across the following series, which I suspect converges to $\ln 2$:
$$\sum_{k=1}^\infty \frac{1}{4^k(2k)}\binom{2k}{k}.$$
I could not derive this series from some of the standard expressions for $\ln 2$. The sum of the first $100 000$ terms agrees with $\ln 2$ only up... | Hint: (or outline -- a lot of details and justifications must be done where there are $(\star)$'s)
*
*for $\lvert x\rvert < \frac{1}{4}$,
\begin{align*}
f(x)&\stackrel{\rm def}{=}\frac{1}{2}\sum_{k=1}^{\infty} \binom{2k}{k}\frac{x^k}{k} = \frac{1}{2}\sum_{k=1}^{\infty} \binom{2k}{k}\int_0^x t^{k-1}dt \\
&\stackrel{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/853229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Nested radicals and n-th roots There are many beautiful infinite radical equations,
some relatively straightforward, some much more subtle:
$$
x = \sqrt{ x \sqrt{ x \sqrt{ x \sqrt{ \cdots } } } }
$$
$$
\sqrt{2} = \sqrt{ 2/2 + \sqrt{ 2/2^2 + \sqrt{ 2/2^4 + + \sqrt{ 2/2^8 + \sqrt{ \cdots}}}}}
$$
$$
3 = \sqrt{1 + 2\sqrt{1... | Here's a nested radical for 4th roots that is not so well known. Given the tetranacci numbers (an analogue of the fibonacci numbers). Let $y$ be the tetranacci constant, or the positive real root of,
$$y^4-y^3-y^2-y-1=0$$
Then,
$$y(3-y) = \sqrt[4]{41-11\sqrt[4]{41-11\sqrt[4]{41-11\sqrt[4]{41-\dots}}}} = 2.06719\dots$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/853813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Proving an identity involving binomial coefficients and fractions I've been trying to prove the following formula (for $n > 1$ natural, $a, b$ non-zero reals), but I don't know where to start.
$$\sum_{j=1}^n \binom{n-1}{j-1} \left( \frac{a-j+1}{b-n+1} \right) \left( \frac{a}{b} \right)^{j-1} \left( \frac{b-a}{b} \right... | Replace $j$ by $j+1$ and $n$ by $n+1$ rewrite as Greg Martin suggested:
\begin{align*}
\sum_{j=0}^n \binom nj \left( \frac{a-j}{b-n} \right) \left( \frac{a}{b} \right)^j \left( \frac{b-a}{b} \right)^{n-j} \tag I
\end{align*}
Let $p=\dfrac{a}{b}$ and $q=1-p$ and consider:
\begin{align*}
\left(q+px\right)^n &= \sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/854050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$ I am currently attempting to prove the following inequality
$\dfrac{a}{b+c}+\dfrac{b}{a+c}+\dfrac{c}{a+b} \geq \dfrac{3}{2}$ for all $ a,b,c>0$
My instinctive plan of attack is to use the AM/GM inequality with $x_1=\dfrac{a}{b+c}$ et... | We can also use rearrangement inequality to prove it.
$\displaystyle\frac a{b+c}+\frac b{a+c}+\frac c{a+b}$
$\displaystyle=\frac12\left(\left(\frac a{b+c}+\frac b{a+c}+\frac c{a+b}\right)+\left(\frac a{b+c}+\frac b{a+c}+\frac c{a+b}\right)\right)$
$\displaystyle\geq\frac12\left(\left(\frac a{a+c}+\frac b{a+b}+\frac c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/855283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 2
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Equation of parabola, tangent at vertex Two tangents on a parabola are $x-y=0$ and $x+y=0$. If $(2,3)$ is the focus of the parabola, then find the equation of tangent at the vertex.
Thanks.
My thoughts:
Can't figure out anything :(
| We are given a pair of perpendicular tangents. The point of intersection of tangents lie on directrix of parabola. Clearly, these tangents intersect at origin. Let the equation of directrix be $y=mx$, then the equation of parabola is
$$(x-2)^2+(y-3)^2=\frac{(y-mx)^2}{1+m^2}$$
Since the parabola is tangent to $y=x$, we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/856771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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inequality method of solution Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$
I've tried first determining when the absolute value will be positive or negative etc, and than giving it the signing in accordance to range it is... | $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 \lt 0 \Rightarrow \\ \left(\left|\frac{x-3}{x+1}\right|\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 \lt 0 $$
$$\Delta=\left(-7\right)^2-4 \cdot 10=49-40=9$$
$$\left|\frac{x-3}{x+1}\right|_{1,2}=\frac{-\left(-7\right) \pm \sqrt{\Delta}}{2}=\frac{7 \pm \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/859209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Finding the derivative of sinus and cosinus. Trigonometric identities How can we see that $$\sin(x+h)-\sin(x)=2\sin\left(\frac h2\right)\cos\left(x+\frac h2\right)$$
How can we see that $$\cos(x+h)-\cos(x)=-2\sin\left(\frac h2\right)\sin\left(x+\frac h2\right)$$
Do these identities have a name?
| You asked for a name: Looks like an application of the reverse of the Prosthaphaeresis identities, or sum-to-product identities.
$$
\sin a − \sin b = 2 \cos\frac{a + b}{2} \sin\frac{a − b}{2} \\
\cos a − \cos b = −2 \sin\frac{a + b}{2} \sin\frac{a − b}{2}
$$
using $a = x + h$ and $b = x$.
Another link here.
I usually u... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Supremum $\sup_x \left\{x-\ln\left(1+\frac{\sin x-\cos x}{2}\right)\right\}$ I would like to know the supremum
$$\sup_x \left\{x-\ln\left(1+\frac{\sin x-\cos x}{2}\right)\right\}$$
when $0\leq x \leq \frac{\pi}{4}$.
After the derivative is easily found that the function is increasing. But, is there a way that does not ... | Let
$$ f(x)=\frac{e^x}{1+\frac{\sin x-\cos x}{2}}. $$
We only have to show $f(x)$ is increasing in $[0,\frac{\pi}{4}]$. In fract, for $0<x_1<x_1+\alpha<\frac{\pi}{4}$ ($\alpha\in(0,\frac{\pi}{4})$),
\begin{eqnarray*}
\frac{f(x_1+\alpha)}{f(x_1)}&=&e^{\alpha}\frac{2+\sin x_1-\cos x_1}{2+\sin (x_1+\alpha)-\cos (x_1+\alph... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve $\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}$ for $x$? Can somebody explain me this one!
$\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}$
My book says the answer is $x_1 = 0$; $x_2 = 6$.
I tried to solve it and got stuck somewhere in:
$4x^2+x+3/(2x-3)(3x-1) = 0$
I can't find my mista... | Note that $$9 - 4x^2 = -(4x^2 - 9) = -(2x+3)(2x-3)$$
$$\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}\frac{2x+1}{2x-3}-\frac{7x\:}{(2x+3)(2x-3)}=1+\frac{x-4}{2x+3}$$
Now, multiply both sides of the equation by $(2x - 3)(2x+ 3)$:
$$(2x+1)(2x+3) - 7x = (2x+3)(2x-3) +(x-4)(2x-3)$$
Simplify by expanding as need... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/865023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Maclaurin series expansion Write the Maclaurin series for: $6\cos{5x^2}$
Find the first five coefficients.
For this question: I repeatedly come to the answer of: $6-75 x^4+\frac{625 x^8}{4}+O(x^9)$
With the coefficients being $6, -75, \frac{625}{4}$ and $0$. These answers are wrong though.
Any suggestions on how to ... | $$
\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \ldots
$$
Thus,
$$
6\cos(5x^2) = 6\biggl[1 - \frac{(5x^2)^2}{2} + \frac{(5x^2)^4}{24} - \frac{(5x^2)^6}{720} + \frac{(5x^2)^8}{40320} - \ldots \biggr]
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/867492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Prove $\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$ I have in trouble for evaluating following integral
$$\int_0^{\infty} \left(\sqrt{1+x^{4}}-x^{2}\right)\ dx=\frac{\Gamma^{2}\left(\frac{1}{4}\right)}{6\sqrt{\pi}}$$
It seems really easy, but I don't kno... | Let $x = \sqrt{\tan \theta}$. Note that, $x^4 = \tan^2 \theta$ and $dx = \dfrac{\sec^2\theta d\theta}{2\sqrt{\tan\theta}}$. Like this,
$$
I = \int_{0}^{\infty}( \sqrt{1 + x^4} - x^2)dx = \int_{0}^{\pi/2} \dfrac{(\sec \theta + \tan \theta)\sec^2\theta d\theta}{2\sqrt{\tan \theta}} = \dfrac{1}{2}\int_{0}^{\pi/2} \dfrac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/867676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 3,
"answer_id": 0
} |
The area of circle The question is to prove that area of a circle with radius $r$ is $\pi r^2$ using integral. I tried to write $$A=\int\limits_{-r}^{r}2\sqrt{r^2-x^2}\ dx$$
but I don't know what to do next.
| Another interesting solution, not yet mentioned by the other answers, is to use this substitution:
$$
x = r\cos\theta \;\therefore\; \frac{dx}{d\theta} = -r\sin\theta
\;\therefore\; dx = -r\sin\theta \; d\theta
$$
Substituting the integral limits, we have
$$x = -r \implies -r = r\cos\theta \implies \cos\theta = -1 \im... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)
Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer.
I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
| Trial division mod $47$, as per the hint.
Firstly, for positive integer $a$, observe that $$a \equiv \overbrace{a \mod 50}^{\text{reduced residue}}+3\lfloor a/50\rfloor \pmod {47}.$$ This makes taking mod $47$s much easier.
*
*We compute $7^2=49 \equiv 2 \pmod {47}$. So $7^8 \equiv 2^4 = 16 \pmod {47}$.
*We compu... | {
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"url": "https://math.stackexchange.com/questions/868663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Prove this polylogarithmic integral has the stated closed form value Question. Prove the following polylogarithmic integral has the stated value:
$$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$
I was able to arrive at the proposed value for the inte... | Since no answers have been posted, I'll expand on my comment above.
There is a general formula that states $$\sum_{n=1}^{\infty}\frac{H_{n}^{(r)}}{n^{q}}=\zeta(r)\zeta(q)-\frac{(-1)^{r-1}}{(r-1)!}\int_{0}^{1}\frac{\text{Li}_{q}(x) \log^{r-1}(x) }{1-x}dx $$ where $$ H_{n}^{(r)} = \sum_{k=1}^{n} \frac{1}{k^{r}} .$$
A p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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maximum area of a rectangle inscribed in a semi - circle with radius r.
A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum.
My Try:
Let length of the side be $x$,
Then the length of th... | Equation of circle: $x^2+y^2=r^2$
$x = \pm (r^2-y^2) $
Thus, length of rectangle is $x-(-x)=2x$ and height is $y$.
Area $A = 2xy$. Maximizing A is equivalent to maximizing $ A^2$
$A^2 = 4x^2y^2 = 4(r^2-y^2)y^2=4r^2y^2-4y^4$
Let, $f(z)= 4r^2z-4z^2 $
Then, $f'(z) = 4r^2-8z$
Equating, $f'(z)=0$ we get,
$\mathbf{z=\frac12r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 7
} |
Find odd numbers $(o_1,o_2,o_3,o_4)$ such that $o_1^2-o_2^2=2(o_3^2-o_4^2)$ such that $o_1>o_2$ and $o_3>o_4$ I am working on a graph labeling problem and am stuck at the following problem on odd numbers.
Find (all) odd numbers $(o_1,o_2,o_3,o_4)$ such that $o_1^2-o_2^2=2(o_3^2-o_4^2)$ such that $o_1>o_2$ and $o_3>o_4$... | Describing all solutions is quite intricate. However, with composite $n \equiv 3 \pmod 8$ where all prime factors of $n$ are $1,3 \pmod 8,$ then there are multiple expressions as $u^2 + 2 v^2;$ with two solutions, you can arrange in your pattern. The predictable kind are illustrated below: if $n$ is the product of $r$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Value of $\cos^2\alpha-\sin^2\alpha$ My problem is from Israel Gelfand's Trigonometry textbook.
Page 48. Exercise 8: b) If $\tan\alpha=r$, write an expression in terms of $r$ that represents the value of $\cos^2\alpha-\sin^2\alpha$.
The attempt at a solution:
Well I solved a) which was similar, question was: a) If $\ta... | My Approach:
$\tan \alpha=r$
$\implies \tan^2 \alpha=r^2$
$\implies \large \frac{\sin^2 \alpha}{\cos^2 \alpha}=r^2$
Apply Componendo and Dividendo
$\implies \large \frac{\sin^2 \alpha+\cos^2 \alpha}{\sin^2 \alpha-\cos^2 \alpha}=\large \frac{r^2+1}{r^2-1}$
$\implies \large \frac{1}{\sin^2 \alpha - \cos ^2 \alpha}=\large... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/873303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
} |
If the circle $x^2+y^2+4x+22y+c=0$ bisects the circmuference of the circle $x^2+y^2-2x+8y-d=0$ the... Problem :
If the circle $x^2+y^2+4x+22y+c=0$ bisects the circmuference of the circle $x^2+y^2-2x+8y-d=0$ then c +d equals
(a) 60
(b) 50
(c) 40
(d) 30
Solution :
Equation of common chord of the circles is given ... | you had an error:
$$S-S'=4x+22y+c-(-2x+8y-d)\\=6x+14y+c+d=0$$
so
$$c+d=-6x-14y$$
because $S$ besects $S'$, so the center of $S'$ is on the line $S-S'$. The center is $(1,-4)$, so
$$c+d=-6*1-14*(-4)=50$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/873506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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