Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Which solution of this integral $\int{\frac{x^2+1}{x^4-x^2+1}}dx$ is correct? $$\begin{align}
\int{\frac{x^2+1}{x^4-x^2+1}}dx&=\int{\frac{1+\frac{1}{x^2}}{x^2-1+\frac{1}{x^2}}}dx\\
&=\int{\frac{1+\frac{1}{x^2}}{(x-\frac{1}{x})^2+1}}dx\\
&=\int{\frac{1}{u^2+1}}du \quad(u=x-\frac{1}{x})\\
&=\arctan(u)+C\\
&=\arctan(x-\fr... | Your answer is$$f(x)=\arctan\left(x-\frac1x\right).$$If you differentiate it, you will get indeed$$\dfrac{x^2+1}{x^4-x^2+1}.\tag1$$But $f$ is undefined at $0$. Besides, $\lim_{x\to0^+}f(x)\ne\lim_{x\to0^-}f(x)$.
On the other hand, if$$g(x)=\arctan\left(\frac x{1-x^2}\right),$$then, again, if you differentiate it, you w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4227462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Average value of $\frac{x'A^2x}{x'A^3x}$ over surface of $n$-dimensional sphere Suppose $A$ is a diagonal matrix with eigenvalues $1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n}$ and $x$ is drawn from standard Gaussian in $n$ dimensions. In numerical simulations, the following quantity seems to converge to $2$ as $n\righ... | Not an answer, but wanted to point out an interesting result leading to an identity with the zeta functions. Let $A = \sum_{k\ge 1} X_k^2/k^2$ and $B = \sum_{k \ge 1}X_k^2/k^3$. Then a Taylor series expansion of $f(A, B)$ around $(\mu_A, \mu_B)=\mathbb{E}[(A, B)] = (\zeta(2), \zeta(3))$ is
\begin{align*}
\frac{A}{B} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4228308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
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How can one solve this without expanding. \begin{array}{l}
\text { If } a+b+c=1, a b+b c+c a=2 \\
\text { and } a b c=3 \text {. What is the value } \\
\text { of } a^{4}+b^{4}+c^{4} \text { ? }
\end{array}
This can be solved by expanding but is there any easy alternative method ?
Here is how I solved :
\begin{array}{l... | $a,b,c$ are roots of $x^3 - x^2 + 2x-3 = 0$
$$a^2+b^2+c^2=(a+b+c)^2 - 2(ab+bc+ac) = 1-2(2)=-3$$
Since
$$x^3 = x^2 -2x+3$$
$$a^3+b^3+c^3 = (a^2+b^2+c^2) - 2(a+b+c) + 9 = -3-2(1)+9=4$$
Since
$$x^4 = x^3 -2x^2+3x$$
$$a^4+b^4+c^4 = 4-2(-3)+3(1)=13$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4229782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Chapter 2 Exercise 1 Pages 83 Linda J. S. Allen 2010 Exercises for Chapter 2 Exercise 1 Pages 83
textbook: An Introduction to Stochastic Processes with Applications to Biology 2nd Edition Linda J. S. Allen 2010
Link to the textbook
My attempt:
$P=(p_{ij})$ where $p_{ij}=\rm{Prob}\left\{X_{n+1}=i\lvert\ X_{n}=j\right\}... | $P= \begin{pmatrix}
p_{11} & p_{12} & p_{13} \\
p_{21} & p_{22} & p_{23} \\
p_{31} & p_{32} & p_{33}
\end{pmatrix}= \begin{pmatrix}
a & 0 & 0 \\
\frac{1}{2} & 0 & \frac{1}{3} \\
0 & b & c
\end{pmatrix}$
Since $\forall j\in [\![1,3]\!];\; \sum_{i=1}^{3}p_{ij}=1$, then it suffices to solve the following system of equat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4230414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Minimum and maximum points using Lagrange's method F(x,y)=x+2y, G(x,y)=x^2+y^2-1
I want to find the minimum and maximum points using lagrange's method, I tried but I always get lost, I could use a little bit help.
Thank you.
| Define,
$L(x,y, \lambda) = F(x,y) + \lambda G(x,y) = x + 2 y + \lambda (x^2 + y^2 - 1) $
Then, at the extreme points,
$\dfrac{\partial L}{\partial x} = 1 + \lambda (2 x) = 0 $
$\dfrac{\partial L}{\partial y} = 2 + \lambda (2 y) = 0 $
$\dfrac{\partial L}{\partial \lambda} = x^2 + y^2 - 1 = 0$
From the first two equation... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Computing local and global minima on Wolfram Alpha Consider $\sqrt{x^2+y^2}+2\sqrt{x^2+y^2-2x+1}+\sqrt{x^2+y^2-6x-8y+25}$. I need to find global or local minima of this function, but Wolfram Alpha doesn't seem to find one; the answer is that $1 + 2\sqrt{5}$ is its global minimum
Am I doing something wrong? I use this i... | Using FindMinimum[Sqrt[x^2+y^2]+2Sqrt[x^2+y^2-2x+1]+Sqrt[x^2+y^2-6x-8y+25],{x,y}] locally, I found the following minimum solutions. The output includes the already mentioned solution $5.47214=1+2\sqrt{5}$ and concretely looks as follows:
$$\{5.47214,\{x\to 1,y\to -7.441589669722822*10^{-10}\}\}$$
Hope it helps you to p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$ Let $a,b,c>0$:
Prove that: $S=\frac{a^2}{(2a+b)(2a+c)}+\frac{b^2}{(2b+c)(2b+a)}+\frac{c^2}{(2c+a)(2c+b)} \leq \frac{1}{3}$
My solution:
We have: $\left[\begin{matrix}\frac{1}{x}+\frac{1}{y} \geq \frac{4}{x+y} \\\... | Let $a\geq b\geq c$.
Thus, $$\frac{1}{3}-\sum_{cyc}\frac{a^2}{(2a+b)(2a+c)}=\sum_{cyc}\left(\frac{a}{3(a+b+c)}-\frac{a^2}{(2a+b)(2a+c)}\right)=$$
$$=\frac{1}{3(a+b+c)}\sum_{cyc}\frac{a(a-b)(a-c)}{(2a+b)(2a+c)}\geq$$
$$\geq\frac{1}{3(a+b+c)}\left(\frac{a(a-b)(a-c)}{(2a+b)(2a+c)}-\frac{b(a-b)(b-c)}{(2b+a)(2b+c)}\right)\g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4234317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Remainder Theorem Technique
Determine the remainder when $(x^4-1)(x^2-1)$ is divided by $1 + x + x^2$ (HMMT 2000, Guts Round)
A. Write the division in the form:
$$(x^4-1)(x^2-1)= (1 + x + x^2)Q(x) + R(x)$$
B. Multiply both sides by $x-1$:
$$(x-1)(x^4-1)(x^2-1)= (x^3-1)Q(x) + R(x)(x-1)$$
C. Substitute $x^3=1,x\neq1$, ... | Alternate method:
$(x^4 - 1)(x^2 -1)$ has a degree of six, we can write:
$$(x^4 -1)(x^2-1)=Q(x) \left[ 1+x +x^2 \right] + R(x) $$
Evaluate (1) at $\{ \omega, \omega^2 \}$:
$$ \begin{align} R( \omega) &= ( \omega-1)(\omega^2 - 1)= ( \omega -1)( \frac{1}{\omega}-1) \\
R( \omega^2) &= (\omega^2 -1)(\omega-1) = ( \omega... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4237066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $x^3+\frac{1}{x^3}$ and $x^4+\frac{1}{x^4}$ are rational numbers. Show that $x+\frac{1}{x}$ is rational.
$x^3+\dfrac{1}{x^3}$ and $x^4+\dfrac{1}{x^4}$ are rational numbers. Prove that $x+\dfrac{1}{x}$ is rational number.
My solution:
$x^3+\dfrac{1}{x^3}=\left(x+\dfrac{1}{x}\right)\left(x^2+\dfrac{1}{x^2}-1\right... | Let $a_k=x^k+\frac1{x^k}$， we have
$\begin{cases}a_1a_4=a_5+a_3\\a_2a_3=a_5+a_1\\a_1a_3=a_4+a_2\end{cases}$
By writing $a_3=u,a_4=v$, we could rewrite the fomula as
$\begin{cases}va_1+0a_2-a_5=u\\a_1-ua_2+a_5=0\\ua_1-a_2+0a_5=v\end{cases}$
Solving the linear equation of $a_1,a_2,a_5$ we could get they're all rational n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4237191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 0
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Respective chances that each of $3$ dice is loaded Here's a question from my probability textbook:
There are three dice A, B, C, two of which are true and one is loaded so that in twelve throws it turns up six $3$ times, ace once, and each of the other faces twice. Each of the dice is thrown three times and A turns up... | A calculation such as
$${{\left({3\over{12}}\right)^2\left({1\over12}\right)}\over{\left({3\over{12}}\right)^2\left({1\over12}\right) + 2\left({1\over6}\right)^3}} = {9\over{25}} = {{63}\over{175}}
$$
makes sense when you only roll $A$, it comes up $6,6,1$, and you want to know the probability that $A$ is loaded. Forma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4237497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proving divisibility by $9$ for $10^n + 3 \times 4^{n+2} + 5$. I am trying to prove that for all $n$, $9$ divides $10^n + 3 \times 4^{n+2} + 5$. I first tried induction on $n$, but couldn't get anywhere with the induction step. I then tried to use modular arithmetic. The furthest I could get was:
As $10 \equiv 1 \mod ... | Method 1: without using induction
$10^n+3. 4^{n+2}+5=(1+9)^n+3(1+3)^{n+2}+5$
Which after applying binomial expansion, takes the form
$1+3(1+3(n+2))+5+9k$ for some positive integer k. This reduces to $9(n+3)+9k$. Hence, $9$ divides the expression for all $n$.
Method 2: Using induction
Let $P(n)$ denote the statement tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4237927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Why does Solving system of quadratic equations gives extra roots? Consider these system of Equations
\begin{align*}
\begin{cases}
x^2+4x+4=0\\\\
x^2+5x+6=0
\end{cases}
\end{align*}
For solving them
We have
Method 1-
Subtract both equations
So
$-x-2=0$
Hence,
$x=-2$
Method-2
Add both equations
$2x^2+9x+10=0$
After apply... | HINT
You can factor both polynomials according to your preferred method in order to obtain:
\begin{align*}
\begin{cases}
x^{2} + 4x + 4 = 0\\\\
x^{2} + 5x + 6 = 0
\end{cases} \Longleftrightarrow
\begin{cases}
(x+2)^{2} = 0\\\\
(x+2)(x+3) = 0
\end{cases}
\end{align*}
Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4238314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Picking two numbers - Calculate the probability We take randomly two numbers without replcement from a box that contains the numbers $1,2,\ldots , 10$.
Calculate the probability of the following events :
a) A = their sum is $11$
b) B = their product is even
c) C = the smaller one is $4$ or $5$
$$$$
I have done the foll... | For $a)$, please note that we could choose any of the $10$ numbers as the first number. Then there is exactly $1$ second number out of remaining $9$ that gives sum of $11$.
So the probability is $\frac{1}{9}$.
In the way you wrote, you should have also counted $10 + 1, 9 + 1$ etc. and the answer should have been $\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4238589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding three unknowns from three equations Let $a$,$b$ and $c$ be three positive real numbers such that
$$\begin{cases}3a^2+3ab+b^2&=&75\\
b^2+3c^2&=&27\\c^2+ca+a^2&=&16\end{cases}$$
Find the value of $ ab+2bc+3ca$.
My attempt: I observed that $3 . 16+27=75$. Then on replacing $16$ by $c^2+ca+a^2$, $27$ by $b^2+3c^2... |
Please note that
$ (\sqrt3 a)^2 + b^2 - 2 (\sqrt3 a) b \cos 150^0 = 3a^2 + b^2 + 3ab = 75$
$b^2 + (\sqrt3 c)^2 - 2 b (\sqrt3 c) \cos 90^0 = b^2 + 3c^2 = 27$
$ (\sqrt3 a)^2 + (\sqrt3 c)^2 - 2 (\sqrt3 a) (\sqrt3 c) \cos 120^0 = 3a^2 + 3 c^2 + 3 a c = 48$
Angles add to $360^0$ so there is a point $O$ inside $\triangle PQ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4241459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Integration of $\sin^2x$ without using double angle identity of $\cos 2x$ I want to integrate $\sin^2x$ without using the double angle identity of $\cos2x$.
Here's what I tried:
$$
\int \sin^2x dx
= \int \sin x \tan x \cos x dx
$$
Let $u = \sin x$ => $du = \cos x dx$
And if $u = \sin x$, $\tan x = \frac{u}{\sqrt{1-u^2}... | You can avoid using the double angle formula by integrating by parts $$\int\sin^2x\,\mathrm{d}x = -\cos x \sin x + \int \cos^2 x \,\mathrm{d}x = -\cos x \sin x + x - \int \sin^2 x \,\mathrm{d}x. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4244411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
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Does Diophantine equation $1+n+n^2+\dots+n^k=2m^2$ have a solution for $n,k \geq 2$? When studying properties of perfect numbers (specifically this post), I ran into the Diophantine equation
$$
1+n+n^2+\dots+n^k=2m^2, n\geq 2, k \geq 2.
$$
Searching in range $n \leq 10^6$, $k \leq 10^2$ yields no solution. So I wonder ... | Suppose we have $(n,s,x,y)$ with $n,s>1$ odd
$$n^s+1=2x^2\quad\text{and}\quad \frac{n^s-1}{n-1}=y^2,$$
then by a result of Ljunggren, we must have $n=3,s=5$, which gives $x^2=122$. Hence, there are no integral solutions to the system of equations in Theorem $1$ (see below). We conclude that there are no solutions in th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4247808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Problem based on area projection in 3-D geometry Considering a Quadrilateral $ABCD$ where $A(0,0,0), B(2,0,2), C(2,2\sqrt 2,2), D(0,2\sqrt2,0)$. Basically I have to find the Area of projection of quadrilateral $ABCD$ on the plane $x+y-z=3$.
I have tried to first find the projection of the points $A,B,C,D$ individually ... | As indicated by Intelligenti pauca in the above comments, the best way to go is to find the area of the quadrilateral, then multiply the area found by the cosine of the angle between the two planes which is the same angle between the normals to the planes (or its supplement).
$\begin{equation} \begin{split}
\text{Area}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4250845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Calculating improper intergal $\displaystyle \int^a_0 \frac{1}{\sqrt{a^2-x^2}} dx$ I tried to solve the improper integral below, but the answer was $\displaystyle \frac{\pi}{2}$. What is wrong with the solution?
$\displaystyle \int^a_0 \frac{1}{\sqrt{a^2-x^2}} dx
= \displaystyle \lim_{p \to a-0} \int^p_0 \frac{1}{\sqr... | Be careful about the change of variable. You need to complete the diferential and change the integration limits. Here $u=\frac{x}{a}$ and $du=\frac{dx}{a}$, then
$\displaystyle \int^a_0 \frac{1}{\sqrt{a^2-x^2}} dx
= \displaystyle \int^a_0 \frac{1}{\sqrt{1-(\frac{x}{a})^2}} \frac{dx}{a} \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4252636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Finding a straight line that makes a given angle with a given straight line Question:
Find the equation of the straight line that passes through $(6,7)$ and makes an angle $45^{\circ}$ with the straight line $3x+4y=11$.
My solution (if you want, you can skip to the bottom):
Manipulating the given equation to get it to ... | I don't see any reason why the clockwise angle would be invalid.
In some contexts, such as trigonometry, we, by convention, write the angle as the counterclockwise angle from the positive $x$-axis. In others, such as true compass bearings, the angle is, again by convention, measured clockwise (from North).
So for this ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Show that the product of the lengths of the perpendiculars drawn from two points to a straight line is $b^2$ Question:
Show that the product of the lengths of the perpendiculars drawn from the points $(\pm c,0)$ to the straight line $bx\cos\theta+ay\sin\theta-ab=0$ is $b^2$ when $a^2=b^2+c^2$.
My attempt:
Let, the len... | Your answer is correct. Note that the absolute value of a real number $x$ is defined by
$$|x|=\begin{cases}x,\space\text{if}x\geq0 \\ -x,\space\text{if} x<0\end{cases}$$
and since $b^2\geq 0$ for all $b\in\mathbb R$ we have $|b^2|=b^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4253565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How can I derive $~\text{opposite}\cdot\sin^{}\left(\theta_{}\right)+\text{adjacent}\cdot\cos^{}\left(\theta_{}\right)=\tan^{}\left(\theta_{}\right)$ Given the below equation .
$$ b \cos^{}\left(\theta_{} \right) = a \cdot \sin^{}\left(\theta_{} \right) $$
I have to derive the below equation .
$$ b \sin^{}\left(\the... |
The diagram makes it easier. It is consistent with your given conditions with $\frac ba = \tan \theta$. Note that $c = \sqrt{a^2 +b^2}$ (Pythagoras).
From the diagram, $\sin \theta = \frac bc, \cos \theta = \frac ac$.
So $b\sin \theta + a\cos \theta = \frac {b^2}c + \frac {a^2} c = \frac {c^2}c = c = \sqrt{a^2 + b^2}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4254289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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What is the tangent to this equation at origin? $$x(x^2 + y^2) = a(x^2 - y^2)$$
I was trying to find the equation of the tangent as $$(Y-0) = \frac{dy}{dx}(X - 0)$$ where
$$\frac{dy}{dx} = \frac{2ax-3x^2-y^2}{2(x+a)y}$$
So here putting the values of $(x,y)$ as $(0,0)$ makes the derivative non existent.
I am stuck at th... |
The appearance of an indeterminate implicit derivative at a point on a curve is often an indication of a self-intersection. This is a folium, the best-known example being that of Descartes $ \ ( x^3 + y^3 - 3axy \ = \ 0 ) \ , \ $ which looks like this one rotated counter-clockwise by $ \ 45º \ \ . $ We see that $ \ ... | {
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"url": "https://math.stackexchange.com/questions/4254761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
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Is this proof that $2^{23} \equiv 1 \bmod{47}$ correct? By Fermat's little theorem, we have that
$$
2^{46} \equiv 1 \pmod{47}
$$
By writing $2^{23}$ as $\left( 2^{46}\right)^{\frac{1}{2}}$ and knowing that
$$
(a^b) \bmod c = ((a \bmod c)^b)\bmod c
$$
we can conclude that
$$
\left( 2^{46}\right)^{\frac{1}{2}} \bmod{47} ... | No, that reasoning is not correct. You are saying: As $2^{46}\equiv 1\bmod 47$ then $2^{23}$ is a square root of $1\bmod 47$. That is true, but there are two possible roots: $\pm1$. So you have either $2^{23}=1$ or $2^{23}=-1$.
Instead you might evaluate $2^{23}\bmod 47$ by square and multiply:
$$ 2^{23} = 2\cdot 2^{2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Probability of One Event Less Than Probability of Second Event I am having a bit of trouble proving some cases when one probability is smaller than the other probability for all positive integers $a, b$, so some suggestions would be appreciated.
Here is the problem:
For all $a, b \in \mathbb{Z}^+$, if $P(A) = \dfrac{a... | I tried to make $P(A)$ and $P(B)$ comparable to avoid case decisions.
\begin{align}
&P(A)=\frac{(a+b)^2-2ab-a-b}{(a+b)^2-a-b}=1-\frac{2ab}{(a+b)^2-a-b}=1-f(a,b)&&\\ \\
& P(B)=\frac{(a+b)^2-2ab}{(a+b)^2}=1-\frac{2ab}{(a+b)^2}=1-g(a,b)&&
\end{align}
Then we start by comparing the denominators.
\begin{align}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4257779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Proving $\frac{2\sin x+\sin 2x}{2\sin x-\sin 2x}=\csc^2x+2\csc x \cot x+\cot^2x$
Prove
$$\dfrac{2\sin x+\sin 2x}{2\sin x-\sin 2x}=\csc^2x+2\csc x \cot x+\cot^2x$$
Proving right hand side to left hand side:
$$\begin{align}\csc^2x+2\csc x \cot x+\cot^2x
&= \frac{1}{\sin^2x}+\dfrac{2\cos x}{\sin^2x}+\dfrac{\cos^2x}{\si... | $$\dfrac{2\sin x+\sin 2x}{2\sin x-\sin 2x}=$$
$$ \frac {2\sin x +2\sin x \cos x }{2\sin x -2\sin x \cos x}=$$
$$\frac {1 + \cos x }{1 -\cos x}=$$
$$\frac {(1 + \cos x)^2 }{1 -\cos^2 x}=$$
$$\frac { 1+2\cos x +\cos ^2 x}{\sin ^2 x}=$$
$$\csc^2x+2\csc x \cot x+\cot^2x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4259836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Question about the proof of $\lim_{x\to a}\frac{x^{n}-a^{n}}{x-a}=na^{n-1}$ Proof:
Let, $x-a=h\implies x=a+h$
$$\lim_{x\to a}\frac{x^{n}-a^{n}}{x-a}$$
$$\lim_{h\to 0}\frac{(a+h)^{n}-a^{n}}{h}$$
$$\lim_{h\to 0}\frac{a^n(1+\frac{h}{a})^{n}-a^{n}}{h}$$
Since $h\to0$, $h$ can be supposed to be less than $a$. So, $|\frac{h}... | Another way.
Use
$x^n-a^n
=(x-a)\sum_{k=0}^{n-1}x^ka^{n-1-k}
$.
Then,
if
$0 < a-c < x < a+c$,
$\begin{array}\\
\dfrac{x^n-a^n}{x-a}
&=\sum_{k=0}^{n-1}x^ka^{n-1-k}\\
\text{so}\\
\dfrac{x^n-a^n}{x-a}
&\lt\sum_{k=0}^{n-1}(a+c)^ka^{n-1-k}\\
&\lt \sum_{k=0}^{n-1}(a+c)^{n-1}\\
&=n(a+c)^{n-1}\\
\text{and}\\
\dfrac{x^n-a^n}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4265134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integral of powers of Bessel function from 0 to infinity Let $J_m(x)$ be the Bessel function of the first kind with order $m$. I was experimenting with the following integral on Wolfram Alpha
$$ \int_0^\infty J_m(x)^4\, dx, $$
and it returns exact value for $m = 1, 2, 3, 4, 5$. Does anyone know if there is an explicit ... | If you consider
$$I_m=2\pi \int_0^\infty \Big[J_m(x)\Big]^4\,dx$$ they are given in terms of Meijer G-functions.
Looking at the first
$$I_0=G_{4,4}^{2,2}\left(1\left|
\begin{array}{c}
1,1,1,1 \\
\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}
\end{array}
\right.\right) \qquad I_1=G_{4,4}^{2,2}\left(1\left|
\begin{arr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4272150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How can I find this limit without L'Hopital and is my current solution correct? Find $$\lim_{x\rightarrow0} \frac{x^4}{1-2\cos x+\cos^2 x}$$
My solution:
Apply L'Hopital
$$=\lim_{x\rightarrow0} \frac{4x^3}{2\sin x-\sin 2x}$$
Apply L'Hopital again
$$=\lim_{x\rightarrow0} \frac{12x^2}{2\cos x-2\cos 2x}$$
Apply L'Hopital ... | Your result looks right.
Without L'Hospital, here is a hint: $1-2\cos x+\cos^2 x=(1-\cos x)^2=4\sin^4\frac{x}{2}$
This follows from double-angle identity using half-angle: $\cos x=\cos^2\frac{x}{2}-\sin^2 \frac{x}{2}= 1-2\sin^2 \frac{x}{2} \implies 2\sin^2 \frac{x}{2}=1-\cos x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4276154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Closed form for infinite sum over unit balls in odd dimensions I stumbled upon the infinite sum
$$
1 + \sum_{k=0}^\infty \frac{(2\pi)^k}{(2k + 1)!!}
= 2 + \frac{2\pi}{3} + \frac{(2\pi)^2}{3\cdot 5} + \frac{(2\pi)^3}{3\cdot 5 \cdot 7} + \dots
$$
(This is actually the sum of the volume of the unit balls in all odd dimens... | Thanks for the comments to @K.defaoite and @Svyatoslav.
From here, we have
$$
\DeclareMathOperator{\erf}{erf}
\sum_{k=0}^\infty \frac{x^k}{(k + 1/2)!}
= \sum_{k=0}^\infty \frac{(2x)^k (2k)!!}{(2k + 1)!}
= \frac{\sqrt{\pi}}{2} \frac{\exp(x)}{\sqrt{x}} \erf{\sqrt{x}},
$$
so
$$
\sum_{k=0}^\infty \frac{(2\pi)^k}{(2k + 1)!... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4278431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to solve this ODE: $y^{(y(x))}(x)=f(x)$? $$\large{\text{Introduction:}}$$
This question will be partly inspired from:
Evaluation of $$y’=x^y,y’=y^x$$
but what if we made the order of an differential equation equal to the function? Imagine that we had the following linear ordinary differential equation using nth d... | Not a complete answer, but a partial answer.
I've asked myself the same question before, but never found a complete solution, so I'd like to share my work on it in the hope that it might help someone. Since nobody has answered yet, I will also share the unfinished one. I hope that's ok.
My Work (the best of it)
Let's a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4279480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
If $\frac{(1+x)^2}{1+x^2}=\frac{13}{37}$, then find the value of $\frac{(1+x)^3}{1+x^3}$
Let $x$ be a real number such that $\frac{(1+x)^2}{1+x^2}=\frac{13}{37}$. What is the value of $\frac{(1+x)^3}{1+x^3}$?
Of course, one way is that to solve for $x$ from the quadratic $37(1+x)^2=13(1+x^2)$ which gives the value $x... | Instead of $ \displaystyle \frac{(1+x)^3}{1+x^3}$, I first find $ \displaystyle \frac{1+x^3}{(1+x)^3}$
$ \displaystyle \frac{1+x^3}{(1+x)^3} = \frac{(1+x) (1+x^2 -x)}{(1+x)^3} = 1 - \frac{3x}{(1+x)^2} \tag1$
(as $x \ne -1$)
Now note that $ \displaystyle \frac{1+x^2}{(1+x)^2} = \frac{37}{13} \implies 1 - \frac{2 x}{(1+x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4282424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
How to get the maximum value of the sum in two sides in two right triangles? The problem is as follows:
The figure from below shows two triangles intersected on point $E$. Assume $AE=3\,m$ and $ED=1\,m$. Find the maximum value of $AB+EC$
The given choices in my book are as follows:
$\begin{array}{ll}
1.&\sqrt{10}\,m... | Any expression in the form $a\sin x+b\cos x$ can be written as $\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}\sin x+\frac{b}{\sqrt{a^2+b^2}}\cos x\right)$. Note that $\left(\frac{a}{\sqrt{a^2+b^2}}\right)^2+\left(\frac{b}{\sqrt{a^2+b^2}}\right)^2=1$ which means that we can interpret these two expressions as cosine and s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4283209",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
In $\triangle ABC$ AB=13,BC=14,CA=15 . D,E,F lie on BC ,CA,DE respectively . AD ,DE,AF perpendicular to BC,AC,BF. segment DF =M/N(coprime) .M+N?
In $\triangle ABC$, $AB=13,BC=14,CA=15$. Points $D,E,F$ lie on $BC ,CA,DE$ respectively such that $AD ,DE,AF$ perpendicular to $BC,AC,BF$. Segment $DF$ =$M/N$(coprime). $M+N$... |
You already obtained $AD = 12, CD = 9, BD = 5$
In $\triangle ACD, CD:AD:AC = 3:4:5$
As $\angle ADE = \angle ACB$, $\triangle ADE \sim \triangle ACD$. So we obtain,
$DE = \dfrac{36}{5}, AE = \dfrac{48}{5}$
Also as $ABDF$ is cyclic, $\angle ABF = \angle ADE = \angle ACB \implies \triangle ABF \sim \triangle ACD$
So, $AF... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4285065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove $\det\begin{bmatrix}A & B\\ B & A\end{bmatrix} = \det(A-B)\det(A+B)$, even when $A$ and $B$ are not commutative. I am aware of the following identity
$\det\begin{bmatrix}A & B\\ C & D\end{bmatrix} = \det(A)\det(D - CA^{-1}B)$
When $A = D$ and $B = C$ and when $AB = BA$ the above identity becomes
$\det\begin{bmatr... | $\begin{pmatrix} A & B \\ B & A \end{pmatrix}\xrightarrow{\text{row1 -= row2}}\begin{pmatrix} A-B & B-A \\ B & A\end{pmatrix}\xrightarrow{\text{col2 += col1}}\begin{pmatrix} A-B & O \\ B & A+B\end{pmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4285189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove $a^2+b^2+c^2=x^2+y^2+z^2$ given that $a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2$ Prove $$a^2+b^2+c^2=x^2+y^2+z^2$$ given that $$a^2+x^2=b^2+y^2=c^2+z^2=\\(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2$$ where $a,b,c,x,y,z \in\mathbb R$
A friend forwarded me this problem which recent... | You are very close. Both your approaches point to the fact that $r$, $s$, $t$ form an equilateral triangle (three points on a circle, with center of mass at the origin). Then, if you use the exponential notation, the points are at $Re^{i\phi}$, $Re^{i\phi+2\pi/3}$, $Re^{i\phi+4\pi/3}$. Then it's easy to show that $$\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4285504",
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"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Minimize of $x^2+y^2$ subject to $x+y \ge 1$
Consider the problem of minimizing $x^2 +y^2$, subject
to $x + y \ge 1$.
Suppose that you start coordinate descent for this problem at $x = 1$ and $y = 0$.
Discuss why coordinate descent will fail.
The primal problem is equivalent to:
minimize $x^2+y^2$ subject to $-x-y \l... | First let's observe from the following figure of the gradient fields for the objective function that the optimum occurs at the boundary, not inside the region bounded by the objective function curve (circle).
Now, by Lagrangian, we have $\nabla f = \lambda \nabla g$, where we have the gradient of the objective functio... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $x - \sqrt{ \frac{2}{x} } = 3 $ then $x-\sqrt{2x} = ?$ Please find the value without calculate the value of $x$. I tried to multipy it by $x$, I tried to square it but I still can't find the solution. I tried to make some equation that my help to solve this:
$$x^{2} + \frac{2}{x}-2\sqrt{2x} =9, $$
$$x^{2} - \sqrt{2x... | \begin{align}
&x - \sqrt{\frac{2}{x}} = 3\\
&x\sqrt{x} - \sqrt{2} = 3\sqrt{x}\\
&\sqrt{x}(x-3)=\sqrt{2}\\
&x(x-3)^2=2\\
&(x-2)(x^2-4x+1)=0
\end{align}
Note that $x=2$ is not a solution, so
$$x^2-4x+1=0 \implies x=2+\sqrt{3} \text{ or } x=2-\sqrt{3}$$
The second is not a solution either, so $x= 2+ \sqrt{3}.$
In that cas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4289510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Urn problem with random halting Consider an urn with $r$ red and $b$ black balls in it. We start drawing balls (without replacement) from it with following rules:
*
*If the drawn ball is red, we stop the draw with probability $p$. Conversely, we continue the draw with probability $1-p$.
*If the drawn ball is black,... | You will draw the first ball with probability $1.$
The probability that you stop after drawing the first ball is
$$ \frac{b + pr}{b + r}. $$
The probability that you draw the second ball is
$$ 1 - \frac{b + pr}{b + r} = \frac{qr}{b + r}.$$
The probability that you draw the $n$th ball,
given that you drew the $(n-1)$st ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4290435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Compute the series $\sum_{n=3}^{\infty} \frac{n^4-1}{n^6}$ Compute the series $$\sum_{n=3}^{\infty} \frac{n^4-1}{n^6}.$$
How do I go about with the index notation, for example to arrange the series instead as $\sum_{n=1}^{\infty}a_n $?
I have tried to simplify the expression as:
$$\begin{align}&\sum_{n=3}^{\infty} \fr... | Just "undo" the extraneous terms.
$$\sum_{n=3}^\infty a_n=\sum_{n=1}^\infty a_n-a_1-a_2.$$
$$\frac{\pi^2}6-\frac54-\frac{\pi^6}{945}+\frac{65}{64}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4290831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Maximizing the cross-sectional area of an isosceles trapezoid with an angle z I have been given the following problem:
A canal must be excavated, which must have a cross section shaped like an isosceles trapezoid. In the cross section, the
bottom line and the two slanting side pieces below the waterline must
together ... | There is another way to do this using the pythagorean theorem. The area of a Trapezoid is half the sum of the bases times the perpendicular height. If we assign $x$ to the $b$ dimension in your diagram then:
$$A = (120+x)(15^2-x^2)^{\frac{1}{2}}$$
Maximum area occurs where: $$\frac{dA}{dx} = 0$$
$$(15^2-x^2)^{\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4292211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$x_1 \leq 4, x_1+x_2 \leq 13, x_1+x_2+x_3 \leq 29, x_1+...+x_4 \leq 54, x_1+...+x_5 \leq 90$. Find the maximum of ...
$x_1 \leq 4, x_1+x_2 \leq 13, x_1+x_2+x_3 \leq 29, x_1+...+x_4 \leq 54, x_1+...+x_5 \leq 90 \text{ for } x_1, ..., x_5 \in R_0^+. \\ \ \\ \text{Find the maximum of } \sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\s... |
Thanks to the hint of Albus Dumbledore:
\begin{align}
&\displaystyle {(10x_5+12x_4+15x_3+20x_2+30x_1)}_{(A)} \bigg(\dfrac 1 {10} + \dfrac 1 {12} + \dfrac 1 {15} + \dfrac 1 {20} + \dfrac 1 {30} \bigg)_{(B)} \geq \Big(\sum \sqrt{x_i}\Big)^2. \\
&A \leq 1200, B = \dfrac 1 3 \\
&\therefore \Big(\sum \sqrt{x_i} \Big)^2 \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4294264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How many negative real roots does the equation $x^3-x^2-3x-9=0$ have? How many negative real roots does the equation $x^3-x^2-3x-9=0$ have ?
My approach :-
f(x)= $x^3-x^2-3x-9$
Using rules of signs, there is 1 sign change , so there can be at most 1 positive real root
f(-x)= $-x^3-x^2+3x-9$
2 sign changes here, indicat... | Given the posted question's precalculus tag, I normally would not have posted the following answer. However, since Siong Thye Goh has given a precalculus answer, I will give an answer that involves derivatives, which is a concept from Calculus (AKA Real Analysis). Note that I am generally ignorant of the precalculus ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4296303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Show that $\cos(2^n)$ diverges Problem:
I have sequence defined as follows:
$$
a_n = \cos(2^n)
$$
I need to show that it diverges.
My progress:
I tried common method here supposing that this sequence has limit $\lim_{n\to\infty}\cos(2^n) = a$ and then trying to get to contradiction with $\lim_{n\to\infty}(\cos(2^{n+1})... | $\textbf{Claim:}$ If $a_{n}$ converges then it converges to either $1$ or $-\frac{1}{2}$.
Proof: Suppose it converges to some $c$ then for each $\epsilon > 0$ $\exists N_{\epsilon} \in \mathbb{N}$ so that if $n \geq N_{\epsilon}$ we have
$$|a_{n+1}-a_{n}| < \epsilon$$
$$|2a_{n}^2-a_{n}-1|<\epsilon$$
This means that $c$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4298571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find all integer solutions of $(a+b+3)^2+2ab=3ab(a+2)(b+2)$ Here is another number theory problem that I am not able to do
Find all integer solutions of $$(a+b+3)^2+2ab=3ab(a+2)(b+2)$$
My attempt$:$
On expanding we get,
$$b^2+4ab+6b+a^2+6a+9=3a^2b^2+6a^2b+6ab^2+12ab$$ or $$-6ab^2+b^2-8ab+6b+a^2+6a+9=3a^2b^2+6a^2b$$ or ... | If $S=a+b$, $P=ab$ we have
$$S^2-6(P-1)S-(3P^2+10P-9)=0\rightarrow$$
$$S_{1,2}=3(P-1)\pm2\sqrt{\Delta}$$
$$\Delta=P(3P-2)$$
Now, we must find $P_i$ such as $\Delta$ be a square: $\Delta=K^2$. That is a difficult. I found only
$P_1=0\rightarrow\Delta=0\rightarrow S=-3$
$P_2=1\rightarrow\Delta=1\rightarrow S=\pm2$
$P_2=9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4299454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find $f(3)$ if $f(f(x))=x^{2}+2$ Let $a,b,f(x),x$ be positive integers such that If $a>b$ then $f(a)>f(b)$ and $f(f(x))=x^{2}+2$ . Find $f(3)$
My approach:
Replacing $x$ with $f(x)$ in the equation gives $f(f(f(x))) = f(x)^2 + 2$, but $f(f(x)) = x^2 + 2$ so $$f(x^2+2) = f(x)^2 + 2$$
how do i proceed after this. Please ... | Note that $f(f(1))=3$ so there is some natural number $n$ such that $f(n)=3$.
If $f(1)>3$ then there could be no solution to $f(n)=3$ so we must have $f(1)\in \{1,2,3\}$.
If $f(1)=3$ then we have $3=f(f(1))=f(3)$, a contradiction.
If $f(1)=1$ then we would have $3=f(f(1))=f(1)$, a contradiction.
Thus $f(1)=2$.
It follo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4300602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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proper formula for this geometric series I'm trying to properly write the expression for this geometric series that converges to $\frac{1}{2}$ for all x > 1. It looks like this:
$x = 2, f(x) = \frac{1}{2}$
$x = 3, f(x) = \frac{1}{6} + \frac{2}{6} = \frac{1}{2}$
$x = 4, f(x) = \frac{1}{24} + \frac{2}{24} + \frac{3}{24} ... | From a "simple solution that meets the requirements" approach, I would recommend using numerators $x-1$ multiplied by each of the numerators from the sum for $f(x-1)$. So for $x=5$ these additional numerators would be $4,8,12,24$ for the sum as $\frac{1+2+3+6+4+8+12+24}{120}$. Breaking down individual terms could be ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4301922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $a, b, c ∈ Z$. Then $ab + ac + bc$ is even if and only if at most one of $a, b$ and $c$ is odd. Is my proof correct? if so, is there something which isn't done nice? I am currently learning proofs on my own and the solutions in my book aren't always complete.
Lemma 1 given: Sum of an odd number of odd numbers is od... | We'll keep in mind that the sum should be even. We'll also use the fact that $ab + bc + ac = b(a + c) + ac$. This will be for the first implication.
If both $a$ and $c$ are even, then $ac$ must be even and $a + c$ must be even. Having an even sum implies that $b(a + c)$ is even and $b$ can be odd or even as desired.
If... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4306016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve $3\sec(x)-2\cot(x)>0$ (trigonometric inequality)? I am struggling to find the solution, here is what I already tried (it's for a pre-calculus class so no calculus):
$3\sec(x)-2\cot(x)=0 \rightarrow 3(\frac{1}{\cos(x)})=2\left(\frac{\cos(x)}{\sin(x)}\right) \rightarrow 3\sin(x)=2\cos(x)^2 \rightarrow 3\sqrt... | $ 3 \sec x - 2 \cot x \gt 0 $
Implies
$ \dfrac{3}{\cos x} \gt 2 \dfrac{\cos x }{\sin x} $
Multiply through by $\sin^2 x \cos^2 x $
$ 3 \cos x \sin^2 x \gt 2 \sin x \cos^3 x $
Hence,
$ \cos x \sin x ( 3 \sin x - 2 \cos^2 x ) \gt 0 $
But $\cos^2 x = 1 - \sin^2 x $, so
$ \cos x \sin x (3 \sin x + 2 \sin^2 x - 2 ) \gt 0 $
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4310998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Number of real solutions of $\begin{array}{r} {\left[\frac{2 x+1}{3}\right]+\left[\frac{4 x+5}{6}\right]} =\frac{3 x-1}{2} \end{array}$ Solve for $x \in \mathbb{R}$ $$\begin{array}{r}
{\left[\frac{2 x+1}{3}\right]+\left[\frac{4 x+5}{6}\right]}
=\frac{3 x-1}{2}
\end{array}$$ where $[x]$ denotes greatest integer less th... | The LHS of this equation is an integer $\,n=\overbrace{\left\lfloor\dfrac{2x+1}3\right\rfloor}^\text{integer}+\overbrace{\left\lfloor\dfrac{4x+5}6\right\rfloor}^\text{integer}=\dfrac{3x-1}2$
We can express $x=\dfrac{2n+1}{3}$ and report in the equation to get $\dfrac 49n+\cdots$ inside the floor values.
This motivates ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4311152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\sum_{k=0}^\infty[\frac{n+2^k}{2^{k+1}}] = ?$ (IMO 1968)
For every $ n \in \mathbb{N} $ evaluate the sum $ \displaystyle \sum_{k=0}^\infty \left[ \dfrac{n+2^k}{2^{k+1}} \right]$ ($[x]$ denotes the greatest integer not exceeding $x$)
This was IMO 1968, 6th problem.
This is a very interesting question I wanted to shar... |
For every $ n \in \mathbb{N} $ evaluate the sum $ \displaystyle \sum_{k=0}^\infty \left[ \dfrac{n+2^k}{2^{k+1}} \right] $
Now,
$\displaystyle \sum_{k=0}^\infty \left[ \dfrac{n+2^k}{2^{k+1}} \right] = \left[ \dfrac{n+1}{2} \right]+ \left[ \dfrac{n+2}{4}\right] + \left[ \dfrac{n+4}{8}\right]+\cdots+\left[ \dfrac{n+2^k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4313244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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How to find the solution of $x^2\equiv 25\pmod{32}$? I am trying to find square root of $57$ modulo $32\times 49$. For that I need to find the solutions of $x^2\equiv 57\pmod{32}$ and $x^2\equiv 57\pmod{49}$ which are $x^2\equiv 25\pmod{32}$ and $x^2\equiv 8 \pmod{49}$. Now I could find the solution of the second one ... | Let $32|x^2-25.$ Then $x$ must be odd so let $x=2n+1.$ Then
$32|(x-5)(x+5)=(2n-4)(2n+6)\;$ iff $\;8|(n-2)(n+3).$
Now one of $n-2,n+3$ is odd and the other is even so it is necessary & sufficient that $8|n-2$ or $8|n+3.$ And the rest is easy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4313816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Epsilon - delta proof $\lim_{x \to \frac{\pi}{4}} \tan(x)=1$ I need to prove the limit using the definition of limit $$\lim_{x\to c}f(x)=L \leftrightarrow \forall \epsilon >0 \hspace{0.5 cm} \exists \delta >0 : 0<|x-c|<\delta \rightarrow |f(x)-L|<\epsilon $$
My attempt
$$|\tan(x)-1|<\epsilon\\ -\epsilon <\tan(x)-1 < \... | Alternative approach:
$\underline{\textbf{Preliminary Results }}$
Result-1
$\displaystyle \lim_{x \to 0} ~\tan(x) = 0.$
Proof:
Attempting to stay within the spirit of the problem,
since the sine function is continuous, I will assume that :
*
*$\displaystyle \lim_{x \to 0} \sin(x) = \sin(0) = 0.$
Edit
If you are giv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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how to solve this linear system of three equations using Cramer's rule? I have a 3-by-3 matrix,
A=$\left [ \begin{matrix}
1 & 2 & 3 \\
1 & 0 & 1 \\
1 & 1 & -1\\
\end{matrix} \right]$
the known terms are (-6, 2, -5), at the right of "=" symbol.
(1) I've calculated the determinant,
(2) I've used Cramer'... | The determinant of$$A_x=\begin{bmatrix}-6&2&3\\2&0&1\\-5&1&-1\end{bmatrix}$$is $6$, which is equal to $\det(A)$. Therefore, $\frac{\det(A_x)}{\det(A)}=1$, which is what you should have got.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4317431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove Cayley-Hamilton theorem I have a exercise in my linear algebra textbook:
Let $c_2\lambda^2+c_1\lambda +c_0=0$ be the characteristic equation for the matrix $$A=\begin{pmatrix}1&3\\3&1\end{pmatrix}$$
Prove that $c_2A^2+c_1A +c_0I=0$
This is Cayley-Hamilton theorem.
My solution:
If a root $\lambda$ exists, then it ... | You have a specific matrix given:
$$
A=\begin{pmatrix}1&3\\3&1\end{pmatrix}.
$$
Its characteristic equation is given by
$$
p(\lambda):=\det(\lambda I - A) = \begin{vmatrix}\lambda-1&-3\\-3&\lambda-1\end{vmatrix}
= (\lambda-1)^2-(-3)^2 = \lambda^2-2\lambda-8.
$$
Now,
$$
A^2=\begin{pmatrix}1&3\\3&1\end{pmatrix}\begin{pma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4320955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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When does this inequality hold? $\sum_{cyc}\frac{a^2+(a+b+d)c}{a^3+3bcd}\ge\frac{(a+b+c+d)^2}{a^3+b^3+c^3+d^3}$
For positive real numbers $a,b,c,d$, prove the inequality $$\frac{a^2+(a+b+d)c}{a^3+3bcd}+\frac{b^2+(a+b+c)d}{b^3+3acd}+\frac{c^2+(b+c+d)a}{c^3+3abd}+\frac{d^2+(a+c+d)b}{d^3+3abc}\ge\frac{(a+b+c+d)^2}{a^3+b^... | Instead of solving LHS we will try to make from RHS
so RHS can be written as
$\frac{( a^2 + b^2 + c^2 + d^2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2cd)}{( a^3 + b^3 + c^3 + d^3 )}$
which can be written as
$\frac{(a^2+c(a+b+d))}{(a^3+b^3+c^3+d^3)}$ + ..... similarly taking a square and 3 other terms
$a^3+b^3+c^3+d^3 \ge a^3+ (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4323556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove that $ \frac{n+1}{4n^2+3}$ is a Cauchy sequence? By definition, I need to show that for any $\epsilon \gt 0$ there exists $N \in \mathbb{N}$ such that for any $m,n \gt N$:
$ \lvert a_n - a_m \rvert \lt \epsilon$
So I write,
$ |\frac{n+1}{4n^2+3} - \frac{m+1}{4m^2+3} |\leq | \frac{n+1}{4n^2+3}| + |\frac{m+1... | It's easier to use the fact that, since $n+1\leqslant2n$ and $4n^2+3\geqslant4n^2$, then$$\frac{n+1}{4n^2+3}\leqslant\frac{2n}{4n^2}=\frac1{2n}.$$Now, use the fact that$$\frac1{2n}<\frac\varepsilon2\iff n>\frac1\varepsilon.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4324086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove fraction inequality proof How can I prove the following inequality?
$$\frac{a^3}{b^3} + \frac{b^3}{c^3} + \frac{c^3}{a^3} \geq \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$$
I've tried the get common denominators but that doesn't really seem to help since although the denominators are the same, the numerators a... | Hints (assuming $\,a,b,c\,$ all have the same sign):
*
*$\displaystyle\frac{a^3}{b^3} + \frac{b^3}{c^3} + \frac{c^3}{a^3} \geq \frac{1}{9}\left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right)^3\,$ by the generalized means inequality;
*$\displaystyle \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \ge 3\,$ by AM-GM.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4330558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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$1^2 - 2^2 + 3^2 - 4^2 + \dots + 1999^2$ using closed form formula for sum of squares of first $n$ natural numbers The question is simple, the solution, not so much
Q.Find the sum of the given expression $1^2- 2^2 + 3^2 - 4^2 + \dots + 1999^2$
My idea is we know $1^2 + 2^2 + 3^2 + 4^2 + \dots + n^2 = \frac{n(n + 1)(2n ... | Looking at your work, if you only subtract the sum of the even terms from the sum of all terms, you are only left with the sum of the odd terms, not the alternating sum you have shown above.
Hint: How many times do I need to subtract a term from a summation to get it to go from + to - ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4335944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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For 2 orthogonal vectors, is tan(a)tan(b)=-1?
This is a part of a physics subproblem I was solving where in the solution they casually mentioned:-
$\begin{align}\\
tan(A) \times tan(B) = -1
\end{align}\\$
Without proving it, and proceeded to solve the complete question. I tried using ASS similarity methods but it yiel... | Use the definition $\tan(x) \triangleq \frac{\sin(x)}{\cos(x)}$ and trigonometric product identities (Google them).
$\begin{align}
\tan(A) \tan(B) &= \frac{\sin(A)}{\cos(A)} \frac{\sin(B)}{\cos(B)}\\
&= \frac{\sin(A) \sin(B)}{\cos(A)\cos(B)}\\
&= \frac{\frac{1}{2} \left(\cos(A - B) - \cos(A + B) \right) }{\frac{1}{2} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4338424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$\int_0^{\frac{\pi}{2}}\ln(\sin^2 x+k^2\cos^2 x)dx$ not by differentiation under the integral? Show that
$$\int_0^{\frac{\pi}{2}}\ln(\sin^2 x+k^2\cos^2 x)dx=\pi\ln \frac{1+k}{2}$$
This is an exercise from Edwards Treatise on Integral Calculus II pg.188.
What solution to this problem can be given ?
In the chapter in Edw... | There is a quite low-level elementary solution. Using some trigonometric identities and change of variable, we obtain
\begin{multline*}
\int\limits_0^{\frac \pi 2} \ln(\sin^2 x + k^2\cos^2x)\,dx = \int\limits_0^{\frac \pi 2} \ln\Big(\frac{1-\cos 2x}2+ k^2\frac{1+\cos 2x}2\Big)\,dx = \\
\frac 12 \int\limits_0^\pi \ln\Bi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Solving the system $x^2+y^2+x+y=12$, $xy+x+y=-7$ I've been trying to solve this system for well over the past hour.$$x^2+y^2+x+y=12$$
$$xy+x+y=-7$$
I've tried declaring $x$ using $y$ ($x=\frac{-7-y}{y+1}$) and solving from there, but I've gotten to $$y^4+3y^3-9y^2-45y+30=0$$ and I don't see how we can get $y$ from here... | $$(x+1)(y+1)=-7+1$$
Let $x+1=a,y+1=b\implies ab=-6$
$$12=x^2+y^2+x+y=(a-1)^2+(b-1)^2+a+b-2=a^2+b^2-(a+b)=(a+b)^2-2(-6)-(a+b)$$
$$\implies(a+b)(a+b-1)=0$$
So, we know $a+b$ and $a,b$
Can you take it home from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4342256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Calculate residue of pole with order 5 I am trying to evaluate the residue of $\frac{(z^6-1)^2}{z^5(2z^4-5z^2+2)}$ at z=0.
Is there a way to do this without having to do the long derivative calculation?
I know we have the formula $Res(f,0)=\lim_{z\to0}\frac{1}{(5-1)!}\frac{\mathrm{d}^{5-1}f}{\mathrm{d}z^{5-1}}z^5f(z)$.... | We will work in the power series ring $\Bbb Q[[z]]$ in the variable $z$ over $\Bbb Q$. Let us denote by $\operatorname{Coeff}_{z^k}\;f$ the coefficient in $z^k$ of some series $f$ in this ring. It may be useful to substitute $w=z^2$, then work in $\Bbb Q[[w]]$, but this is important only to have an easier typing. We ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4342553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Limit point of sequence whose general term is $ \frac{1}{1\cdot n} + \frac{1}{2\cdot(n-1)}+\dots+\frac{1}{n\cdot1}$ I need to find the limit point(s) of the sequence whose general term is given by:
$$a_n = \frac{1}{1\cdot n} + \frac{1}{2\cdot (n-1)}+\dots+\frac{1}{n\cdot 1}$$
My observation:
*
*$\frac{1}{1\cdot n} + ... | The sequence, $a_n$, can be written
$$\begin{align}
a_n&=\sum_{k=1}^{n} \frac{1}{k(n-k+1)}\\\\
&=\frac1{n+1}\sum_{k=1}^{n}\left(\frac1k-\frac1{k-(n+1)}\right)\\\\
&=\frac2{n+1}\sum_{k=1}^n \frac1k
\end{align}$$
It is easy to show that $a_n$ is monotonically decreasing and bounded below by $0$. So $a_n$ converges. Mo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4344136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Convergence of the series with $a_n= \frac{1}{\sqrt{1} \cdot n^2} + \frac{1}{\sqrt{2} \cdot (n-1)^2} +\cdot\cdot\cdot + \frac{1}{\sqrt{n} \cdot 1^2}$ I need to determine the convergence of the series whose general term is given by:
$$a_n= \frac{1}{\sqrt{1} \cdot n^2} + \frac{1}{\sqrt{2} \cdot (n-1)^2} +\cdot\cdot\cdot ... | $S_N= \sum_{n=1}^N \sum_{k=1}^n \frac{1}{\sqrt{k}(n+1-k)^2}$
we have
$1\leq k \leq n\leq N$
So
$S_N=\sum_{k=1}^N \sum_{n=k}^N \frac{1}{\sqrt{k}}\frac{1}{(n+1-k)^2}$
$= \sum_{k=1}^N\frac{1}{\sqrt{k}} \left(\sum_{n=k}^N \frac{1}{(n+1-k)^2}\right)$
But
$\sum_{n=k}^N \frac{1}{(n+1-k)^2}\geq \int_{1}^{N-k+2} \frac{1}{x^2}dx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4344372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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For what values of $c$ will $\sum_{n=1}^{\infty}\left(\frac{c}{n} - \frac{1}{n+1}\right)$ converge? (verify solution) For what values of $c$ will $$\sum_{n=1}^{\infty}\left(\frac{c}{n} - \frac{1}{n+1}\right)$$ converge?
The $n$th partial sum is given by:
\begin{align*}s_n&=c+\frac{c-1}{2}+\frac{c-1}{3}+ ... + \frac{c-1... | You might pave the way differently. We know that if $p>1$ and $$\displaystyle\lim_{n\to\infty}n^pu_n<\infty$$ then $\sum u_n$ converges. This, for the question, is equivalent to have $\displaystyle\lim_{n\to\infty}n^{p-1}(c-1)<\infty$, so it should be $c=1$ if we are looking for the convergence!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4344870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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For complex $z$, find the value of $z$; $z^2 -2z+( 3-4i) =( 6+3i)$ \begin{array}{l}
\boldsymbol{z^{2} -2z+( 3-4i) =( 6+3i)}\\\\
z^{2} -2z+( 3-4i) =( 6+3i)\\
x^{2} -y^{2} +2xyi-2x-2yi-3-7i=0\\
\left( x^{2} -y^{2} -2x-3\right) +i( 2xy-2y-7) =0\\
x^{2} -y^{2} -2x-3=0\rightarrow ( 1)\\
2xy-2y-7=0\rightarrow ( 2)\\
( 2) \ r... | Yo can do this:
\begin{align*} z^2 -2z+( 3-4i) &= 6+3i
\\ z^2 -2z+1+2-4i & = 6+3i
\\ z^2 -2z+1 & = 4+7i
\\ (z-1)^2 & = 4+7i
\\ z-1 & =(4+7i)^{1/2}\end{align*}
Remember that for complex numbers, the square root has two values. And so
$$z_1=\sqrt{65}e^{\arctan(7/4)i/2},\quad z_2=\sqrt{65}e^{(\arctan(7/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4347485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$(x-y)(f(f(x)^2)-f(f(y)^2))=(f(x)-f(y))(f(x)^2-f(y)^2$
Find $f: \mathbb{R} \to \mathbb{R}$ which satisfies:
$f\small(0)\normalsize=0, f\small(1)\normalsize=2015. \\
(x-y)(f\small(f(x)^2\small)\normalsize-f\small(f(y)^2\small)\normalsize)=(f\small(x)\normalsize-f\small(y)\normalsize)(f\small(x)^2\normalsize-f\small(y)^... | Posted on AOPS, and I am posting the arranged solution.
\begin{align}
P(x, y): \; & (x-y)(f(f(x)^2)-f(f(y)^2))=(f(x)-f(y))(f(x)^2-f(y)^2).\\
P(x, 0): \; & xf(f(x)^2)=f(x)^3. \\
\ \\
\therefore \; & yf(f(x)^2)+xf(f(y)^2)=f(x)f(y)(f(x)+f(y)). \text{ (Let this one be $Q(x, y).$)} \\
\ \\
Q(1, 1): \; & f(2015^2)=2015^3. \\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4360525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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What is the Taylor's expansion of $f(x)=\exp(\frac{1}{2}c^2x^2)$ with a constant $c$? What is the Taylor's expansion of $f(x)=\exp(\frac{1}{2}c^2x^2)$ with a constant $c$.
Note that $f'(x)=\exp(\frac{1}{2}c^2x^2)c^2x$ and $f''(x)=\exp(\frac{1}{2}c^2x^2)(c^2x)^2+\exp(\frac{1}{2}c^2x^2)c^2$. Then $f'(0)=0$ and $f''(0)=c^... | Alright, first to simplify the problem you are asking what the Taylor Series expansion of $\exp(ax^2)$ for a constant $a$. In your case, simply take $a=\frac{1}{2}c^2$. But then
$$\exp(x)=\sum_{n=0}^\infty \frac{x^n}{n!}$$
$$\Rightarrow \exp(ax^2)=\sum_{n=0}^\infty \frac{(ax^2)^n}{n!}=\sum_{n=0}^\infty \frac{a^n x^{2n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4360988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which of the two quantities $\sin 28^{\circ}$ and $\tan 21^{\circ}$ is bigger . I have been asked that which of the two quantities $\sin 28^{\circ}$ and $\tan 21^{\circ}$ is bigger without resorting to calculator.
My Attempt:
I tried taking $f(x)$ to be
$f(x)=\sin 4x-\tan 3x$
$f'(x)=4\cos 4x-3\sec^23x=\cos 4x(4-3\sec^2... | We can use some trigonometric identities to help us here. We need to know three things here:
*
*$\sin 2\theta = \frac{2\tan \theta}{1+\tan^2\theta}$
*$\tan 2\theta = \frac{2\tan \theta}{1-\tan^2\theta}$
*$\tan 3\theta = \frac{3\tan\theta-3\tan^3\theta}{1-3\tan^2\theta}$
You were onto something when looking at $\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4361575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
} |
Deriving the formula for $\cos^{-1}x+\cos^{-1}y$ I am trying to prove the following:
$$\cos^{-1}x+\cos^{-1}y=\begin{cases}\cos^{-1}(xy-\sqrt{1-x^2}\sqrt{1-y^2});&-1\le x,y\le1\text{ and }x+y\ge0\\2\pi-cos^{-1}(xy-\sqrt{1-x^2}\sqrt{1-y^2});&-1\le x,y\le1\text{ and }x+y\le0\end{cases}$$
Let $\cos^{-1}x=A, \cos^{-1}y=B.... |
$\cos^{-1}(xy-\sqrt{1-x^2}\sqrt{1-y^2})=\cos^{-1}(\cos(A+B))=\begin{cases}A+B;&0\le A+B\le\pi\implies0\le A,B\le\frac\pi2\\2\pi-(A+B);&\pi\le A+B\le2\pi\implies\frac\pi2\le A,B\le\pi\end{cases}$
$0\leq A+B \leq\pi$
Now, A and B is always greater than $0$,therefore $0\leq A+B$ is always true.
Leaving us with,
$\implie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4365736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to find the formula for the integral $\int_{0}^{\infty} \frac{d x}{\left(x^{2}+1\right)^{n}}$, where $n\in N$? By the generalization in my post,we are going to evaluate the integral $$\int_{0}^{\infty} \frac{d x}{\left(x^{2}+1\right)^{n}},$$
where $n\in N.$
First of all, let us define the integral $$I_n(a)=\int_{0}... | Denote $I_n = \int_0^\infty \frac1{(1+x^2)^n}dx$, and note that
$$\left( \frac x{(1+x^2)^{n-1}} \right)’
= \frac {2(n-1)}{(1+x^2)^{n}}- \frac {2n-3}{(1+x^2)^{n-1}}
$$
Integrate both sides to get the recursion $I_n= \frac{2n-3}{2(n-1)}I_{n-1} $, along with $I_1=\frac\pi2$. Thus
$$I_n = \frac{(2n-3)!(n-1)\pi }{[2^{n-1} (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4365867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
Evaluating $\int\frac{\sec^2(x)}{(4+\tan^2(x))^2}\, dx$ How to solve this integral?$$\int\frac{\sec^2(x)}{(4+\tan^2(x))^2}\, dx$$
I've tried the following:
Starting by substituting $\tan(x) = 2\tan(\theta)\implies \sec^2(x)\ dx= 2\sec^2(\theta)\ d\theta$
$$\implies\int\frac{2\sec^2(\theta)}{(4 + 4\tan^2\theta)^2}\ d\th... | Wolfram|Alpha returns an antiderivative of
$$\int \frac{\sec^2(x)}{\left(4+\tan^2(x)\right)^2} \, dx = \frac{2\sin(2x) - (3\cos(2x)+5) \arctan\left(2\cot(x)\right)}{48\cos(2x) + 80} + C$$
which we can rewrite as
$$-\frac1{16} \arctan\left(2\cot(x)\right) + \frac18 \cdot \frac{\sin(2x)}{3\cos(2x) + 5} + C$$
The antider... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4367610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
evaluate the integral involving $e^{-x}$ Evaluate
$$\int_{\ln 2}^{\ln 4} \frac{e^{-x}}{\sqrt{1-e^{-2x}}}dx$$
I made the $u-$sub
$$u=e^{-x}$$
Then my limits become $1/2$ and $1/4$ and
$$du = - e^{-x} dx$$
so rewriting the integral I have
$$-\int_{1/2}^{1/4}\frac{1}{\sqrt{1-u^2}}du=\int_{1/4}^{1/2}\frac{1}{\sqrt{1-u^2}}d... | Notice that :
$$\int_{1/4}^{1/2} \dfrac{\mathrm{d}u}{\sqrt{1 - u^2}} = -\left[\cos^{-1} u\right]_{1/4}^{1/2} = \cos^{-1} \dfrac{1}{4} - \cos^{-1} \dfrac{1}{2} = \cos^{-1} \dfrac{1}{4} - \dfrac{\pi}{3}$$
But we know that :
$$\cos^{-1} x = \sin^{-1} \sqrt{1 - x^2}$$
then :
$$\int_{1/4}^{1/2} \dfrac{\mathrm{d}u}{\sqrt{1 -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4372569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find all solutions to $\frac{a+b}{b} = \frac{a}{a+b}$ I want to find all solutions to $\frac{a+b}{b} = \frac{a}{a+b}$. I have plugged this into a calculator which tells us that if we solve for $a$, without loss of generality, we obtain
$$a = - \frac{b}{2} \pm \frac{\sqrt{3}}{2}i$$
and vice versa. I am not sure how to s... | You have $(a+b)^2=ab$ or $a^2+ab+b^2=0$. If $a\ne b$, multiply by $(a-b)$, get $a^3-b^3=0$. So $a=b\omega$ where $\omega$ is one of the cube roots of $1$. The possibility $a=b$ does not work, the other two work.
Answer: $b$ is any non-zero number, $a=b\omega$ where $\omega$ is one of the two non-1 cube roots of $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4375139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
n-th root in limits I'm having trouble with this kind of series: $$a_n = \sqrt{n}(\sqrt[7]{n+5}-\sqrt[7]{n-4})$$
I tried to make something like perfect square to simplify those $7$th root junk which is obviously impossible, and as $n$ tends to infinity L'Hopital's rule also can't solve the problem in this form.
The mul... | Rewrite it $$\sqrt{n}\left(\sqrt[7]{n+5} - \sqrt[7]{n-4}\right) = \\ n^{\frac{9}{14}}\left((1+\frac{5}{n})^{\frac{1}{7}}-(1-\frac{4}{n})^{\frac{1}{7}}\right) \\ \approx n^{\frac{9}{14}}\left(1+\dfrac{1}{7}\cdot\dfrac{5}{n} - 1 +\dfrac{1}{7}\cdot\dfrac{4}{n} + o(\dfrac{1}{n})\right) \\ \approx \dfrac{9}{7n^{\frac{5}{14}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4375628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Solving $\overline{z}\cdot|z|\cdot z^5=8\sqrt{2}\left(-\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^8$ I have this equation to solve:
$$\overline{z}\cdot|z|\cdot z^5=8\sqrt{2}\left(-\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^8$$
Since $\overline{z}\cdot z = |z|^2$ and utilizing the de Moivre's formula this can be simpl... | Let $z = A e^{i \theta} $ then
$ A^7 e^{i 4 \theta} = 8 \sqrt{2}(e^{-i \dfrac{3 \pi}{10}} )^8 = 2^{7/2} e^{-i \dfrac{12 \pi}{5}} = 2^{7/2} e^{i \dfrac{8 \pi}{5}} $
Hence, $A = \sqrt{2}$, and $\theta = \dfrac{2\pi}{5}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4378591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Over the field $\mathbb{Z}_2$, does $x^6 + x^3 + 1$ divide $(x^4 + 1)^n + 1$ for some $n \in \mathbb{N}$? I am currently investigating whether the polynomial $x^6 + x^3 + 1 \in \mathbb{Z}_2[x]$ divides $(x^4 + 1)^n + 1$ for some $n \geq 1$.
I think it is impossible, and so far, I've checked in the case where $n$ is a p... | This $n$ exists. Let $F$ be the splitting field of $x^6+x^3+1$ over $\Bbb F_2$. Let $n=|F|-1$. Then for every $y\ne 0$ in $F$ we have $y^n=1=-1$, so $y^n+1=0$. The polynomial $x^6+x^3+1$ has $6$ roots in $F$ and no double roots. Since every root of $x^6+x^3+1$ is a root of $(x^4+1)^n+1$, $x^6+x^3+1$ divides $(x^4+1)^n+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4378954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the number of complex numbers satisfying $|z|=\text{max}\{|z-1|,|z+1|\}$
Find the number of complex numbers satisfying $|z|=\text{max}\{|z-1|,|z+1|\}$
My Attempt: Let $z=x+iy$, so, $$\sqrt{x^2+y^2}=\text{max}\{\sqrt{(x-1)^2+y^2},\sqrt{(x+1)^2+y^2}\}$$
Case I: $\sqrt{x^2+y^2}=\sqrt{(x-1)^2+y^2}\implies \pm x=x... | $|z| = max \{ |z-1|, |z+1|\} \implies |z| \ge |z-1|$ and $|z| \ge |z+1|$ . Summing the two equations, we have $|2z| \ge |z-1| + |z+1|$. But triangle inequality states that $|z-1| + |z+1| \ge |2z|$ so $|z-1| + |z+1| = 2|z|$. Because, $|z|$ is one of $|z-1|$ and $|z+1|$, we then deduce that $|z-1| = |z| = |z+1|$. $z$, $z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4384485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
$ab+ac+bc=2\quad ,\quad\min(10a^2+10b^2+c^2)=?$
If $ab+ac+bc=2$, find minimum value of $10a^2+10b^2+c^2$
$1)3\qquad\qquad2)4\qquad\qquad3)8\qquad\qquad4)10$
I used AM-GM inequality for three variables:
$$ab+ac+bc\ge3(abc)^{\frac23}\quad\Rightarrow\quad 2\ge3(abc)^{\frac23}$$
$$10a^2+10b^2+c^2\ge 3(10abc)^{\frac23}$$
... | Suppose $a,b,c>0$. Let
$$ a=\frac1{\sqrt{10}}r\cos\phi\sin\theta,b=\frac1{\sqrt{10}}r\sin\phi\sin\theta,c=r\cos\theta. $$
Then
$$ 10a^2+10b^2+c^2=r^2$$
and
\begin{eqnarray}
&&ab+ac+bc\\
&=&\frac1{20}r^2\sin(2\phi)\sin^2\theta+\frac1{2\sqrt{10}}r^2\cos\phi\sin(2\theta)+\frac1{2\sqrt{10}}r^2\sin\phi\sin(2\theta)\\
&=&\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4387544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evaluation of $\sqrt[3]{40+11\sqrt{13}}+\sqrt[3]{40-11\sqrt{13}}$
Evaluate
$$\sqrt[3]{40+11\sqrt{13}}+\sqrt[3]{40-11\sqrt{13}}$$
The solution is $5$.
Suppose $\sqrt[3]{40+11\sqrt{13}}=A, \sqrt[3]{40-11\sqrt{13}}=B$
We have
$$A^3+B^3=80, A^3-B^3=22\sqrt{13}$$
Two unknowns, two equations, so we should be able to solve ... | Hint 1
$$A^3+B^3=(A+B)((A+B)^2-3AB)$$
Hint 2
$$AB=3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/4391660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Maximizing the volume of a cuboid with constraints (lagrange) fails? Given is the following (translated) problem:
You have $12$ meters of wire.
Try to build a wireframe model of a cuboid with sidelengths $x, y, z$ and maximize volume $V(x,y,z)=xyz$.
Show that this is the case iff all sidelengths are equal.
My attempt:
... | We have to consider the so-called Bordered Hessian that is the Hessian of the Lagrange function. Here, at the point $(1,1,1,-1)$, it is equal to
$$H_L=\begin{pmatrix}
0 & g_x & g_y& g_z\\
g_x & L_{xx} & L_{xy} & L_{xz} \\
g_y & L_{yx} & L_{yy} & L_{yz} \\
g_z & L_{zx} & L_{zy} & L_{zz}
\end{pmatrix}=\begin{pmatrix}
0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4391783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculate the Arc length of a Polar curve $r=1+\cos(2\theta)$
Given the polar equation, $r=1+\cos(2\theta)$, prove that the length of the curve corresponding to $0 \leq\theta \leq2\pi$ is $8+\frac{4}{\sqrt{3}}\log(2+\sqrt{3})$.
Applying integration to find arc length, $l$, in polar form:
$\int_\alpha^\beta\sqrt{r^2... | $r = 1 + \cos2\theta = 2 \cos^2\theta$
$r_{\theta} = - 4 \cos\theta \sin\theta$
$ \displaystyle l = 4 \int_0^{\pi/2}\sqrt{4\cos^4\theta + 16 \cos^2\theta \sin^2\theta} ~ d\theta$
$ \displaystyle l = 8 \int_0^{\pi/2} \cos\theta \sqrt{1 + 3 \sin^2\theta} ~ d\theta$
Now substitute $t = \sin\theta$. Can you take it from he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4397148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to apply Stolz–Cesàro theorem in this question? $$\lim _{n \rightarrow \infty}\left(\frac{2}{2^{2}-1}\right)^{\frac{1}{2^{n-1}}} \left (\frac{2^{2}}{2^{3}-1}\right)^{\frac{1}{2^{n-2}}} \cdots\left(\frac{2^{n-1}}{2^{n}-1}\right)^{\frac{1}{2}}$$
I don't know how to continue after taking the logarithm of it, after tha... | Note
\begin{eqnarray}
&&\ln\bigg[\left(\frac{2}{2^{2}-1}\right)^{\frac{1}{2^{n-1}}} \left (\frac{2^{2}}{2^{3}-1}\right)^{\frac{1}{2^{n-2}}} \cdots\left(\frac{2^{n-1}}{2^{n}-1}\right)^{\frac{1}{2}}\bigg] \\
&=&\sum_{k=2}^n\frac{1}{2^{n-k+1}}\ln\bigg(\frac{2^{k-1}}{2^{k}-1}\bigg)\\
&=&\frac{\sum_{k=2}^n2^k\ln\bigg(\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4397318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Product of real quadratic roots If product of roots '$p$' of equation $x^2 – 2ax + 8 – a = 0$ lies between its roots, then maximum
integral value of '$p$' is _____
My approach is as follow
$f\left( x \right) = {x^2} - 2ax + 8 - a$
${D^2} = 4{a^2} - 4\left( {8 - a} \right) \ge 0 \Rightarrow {a^2} + a - 8 \ge 0$
${a^2} +... | The roots are
$$a\pm \sqrt{a^2-(8-a)}$$
and the product of the roots is $8-a$, so defining $p=8-a$, we wish to maximize $p$ (among integers) subject to constraints
$$ 8-p-\sqrt{(8-p)^2-p}\leq p\leq 8-p+ \sqrt{(8-p)^2-p}.$$
A quick sketch shows optimally the upper constraint must bind so
$$p= 8-p+ \sqrt{(8-p)^2-p}\impli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4397707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What's the ratio between the segments $\frac{AF.BG}{FG}$ in the figure below? For reference: In the figure below the trapezoid has height $13$ and is inscribed in a circle of radius $15$. Point $E$ is on the minor arc determined by $A$ and $B$, the points $F$ and $G$ intersect with $ED$, $E$C and $AB$. Then the ratio b... | Can you see that $~\triangle AFE \sim \triangle CBE~$ and $~\triangle BGE \sim \triangle DAE~$?
That leads to
$~ \displaystyle \frac{BC}{AF} = \frac{CE}{AE}~, \frac{AD}{BG} = \frac{AE}{GE}$
Multiplying, $\displaystyle \frac{BC \times AD}{AF \times BG} = \frac{CE}{GE} \tag1$
As $ \displaystyle \triangle FEG \sim \tri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4400350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Integral on a rotational solid
Compite the integral $$\int_V\frac{4-z^2}{(x^2+y^2)^3}\mathrm dx\mathrm dy\mathrm dz,$$ where $V$ is the solid enclosed by the paraboloids $x^2+y^2=z,x^2+y^2=2z$ and cones $x^2+y^2=(z-2)^2,x^2+y^2=4(z-2)^2.$
My attempt:
$V$ is a rotational solid with $z$ axis as the axis of the rotation... | Let $D_-=\{(r,z): r\geq 0, z\leq 2, z\leq r^2\leq 2z, (z-2)^2\leq r^2\leq 4(z-2)^2\}$, and consider the following change of variables: $u=z/r^2$ and $v=(z-2)^2/r^2$, then
$$\left|\det\begin{pmatrix}
\frac{\partial u}{\partial r}& \frac{\partial u}{\partial z}\\
\frac{\partial v}{\partial r}& \frac{\partial v}{\partial ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4401924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the inverse of a piecewise function I am struggling with finding the inverse of a piecewise function, namely:
$f(x) = \begin{cases}
x - 1, & 0 \le x < 1\\
2 - x,& 1 < x \le 2 \\
\end{cases} $
For the first case $\\$
$f: y = x - 1$ when $0 \le x < 1 \rightarrow f^{-1}: x = y + 1$ ... | Suppose that $f^{-1}$ there exists, then you have $f(x)=x-1$ when $0\leqslant x<1$ then setting $y=f(x)$ we get $y=x-1$ when $0\leqslant x<1$ then $x=y+1$ when $0\leqslant y+1<1$. Hence $f^{-1}(x)=x+1$ when $-1\leqslant x<0$.Also $f(x)=2-x$ when $0<x\leqslant 2$ then setting $y=f(x)$ we get $y=2-x$ when $1<x\leqslant 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4403034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Asking About Best Upper Bound And lowerBound Find the best lower and upper bounds for
$$\left(\cos A-\sin A\right)\left(\cos B-\sin B\right)\left(\cos C-\sin C\right),$$
$1)~~$ overall acute-angled $\Delta ABC.$
first of all we know that $\cos(A)-\sin(A) = -\sqrt{2}\sin(A - \frac{\pi}{4})$
$\Pi { -\sqrt{2}\sin(A - \fra... | $\left(\cos A-\sin A\right)\left(\cos B-\sin B\right)\left(\cos C-\sin C\right) = -2\sqrt{2} \prod_{cyc} (A - \pi / 4)$.
WLOG, $\pi / 2 > A \geq B \geq C > 0$.
Easy to see $\pi /3 \leq A < \pi/2$, $\pi/4 < B \leq A$, and $0 < C \leq B$.
Clearly, upper bound is achieved when $\sin(C - \pi/4) < 0$, i.e., $0 < C < \pi/4$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4409059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Evaluating $\int_{-\infty}^\infty \frac{\ln{(x^4+x^2+1)}}{x^4+1}dx$ I recently attempted to evaluate the following integral
$$\int_{-\infty}^\infty\frac{\ln{(x^4+x^2+1)}}{x^4+1}dx$$
I started by inserting a parameter, $t$
$$F(t)=\int_{-\infty}^\infty\frac{\ln{(tx^4+x^2+t)}}{x^4+1}dx$$
Where F(0) is the following
$$F(0)... | \begin{align}J&=\int_{-\infty}^\infty \frac{\ln{(x^4+x^2+1)}}{x^4+1}dx\\
&=2\int_{0}^\infty \frac{\ln{(x^4+x^2+1)}}{x^4+1}dx\\
&=4\int_{0}^\infty \frac{\ln x}{1+x^4}dx+\underbrace{\int_{0}^\infty \frac{\left(1+\frac{1}{x^2}\right)\ln\left(\left(x-\frac{1}{x}\right)^2+3\right)}{\left(x-\frac{1}{x}\right)^2+2}dx}_{u=x-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4411366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 2,
"answer_id": 0
} |
Convergence of double summation of divergent and convergent series I am looking at two series
(a)$$\sum_{n = 1}^\infty \dfrac{1}{n^2 (n + 1)^{2p}}\sum_{m = 1}^n (m + 1)^{2p}$$
and
(b)
$$\sum_{n = 1}^\infty \dfrac{1}{(n + 1)^{2p}}\sum_{m = 1}^n \dfrac{(m + 1)^{2p}}{m^2}$$
Assume $p \in (0, 1]$ for both (a) and (b). I wi... | The first series is divergent for all $p>0$, and therefore so is the second, whose terms are larger.
To see this, consider $\alpha>0$, and let's show that $\sum_{k=1}^nk^\alpha\sim Cn^{\alpha+1}$ for some constant $C$.
We have
$$\frac{1}{n^{\alpha+1}}\sum_{k=1}^nk^\alpha=\frac{1}{n}\sum_{k=1}^n\left(\frac kn\right)^\al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4411568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Integrating $\int_{-\infty}^{\infty} \big(\frac{x^2}{x^2 + a}\big)^2 e^{-x^2/2} \, \mathrm{d}x$ I was wondering if it is possible to compute this integral in closed form:
$$
\int_{-\infty}^{\infty} \big(\frac{x^2}{x^2 + a}\big)^2 e^{-x^2/2} \, \mathrm{d}x
$$
I tried making a substitution with $s = x^2$ and also tried e... | This is too long for a comment.
Since, in comments, you already received the answer and a good hint, let me address the more general case of
$$I_n= \int_{-\infty}^{+\infty} \left(\frac{x^2}{x^2+a}\right)^n\, e^{-\frac{x^2}{2}}\,dx=2\int_{0}^{+\infty} \left(\frac{x^2}{x^2+a}\right)^k\, e^{-\frac{x^2}{2}}\,dx$$
The idea ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4413945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to find the probability density function of the random variable $\frac{1}{X}$? How to find the probability density function of the random variable $\frac{1}{X}$?
Let $X$ be a random variable with pdf
$f_X(x)= \begin{cases} 0 ; x \le 0 \\ \frac{1}{2} ; 0 < x \le 1 \\ \frac{1}{2x^2} ; 1 < x < \infty \end{cases}$
I wa... | The PDF of $X$ is given by
$$
f_X(x) = \left\{ \begin{array}{ccc}
0 & \mbox{if} & x \leq 0 \\[2mm]
{1 \over 2} & \mbox{if} & 0 < x < 1 \\[2mm]
{1 \over 2 x^2} & \mbox{if} & 1 < x < \infty \\[2mm]
\end{array} \right. \tag{1}
$$
Thus, $X$ is a positive random variable.
Since $Y = {1 \over X}$, it is immediate that $Y$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4419761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Can we prove the inequality without opening the parentheses? $(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$
Let, $x,y,z>0$ such that $ xyz=1$, then prove that
$$(x+y+z)(xy+yz+xz)(x^2+y^2+z^2)≥6(x^2+y^2+z^2)+3(xy+yz+xz)$$
I tried to use the following inequalities:
$$x^2+y^2+z^2≥xy+yz+xz$$
and The Cauchy–S... | Increase the right hand side to $9(x^2+y^2+z^2)$ by switching $xy+yz+zx$ to $x^2+y^2+z^2$ (Cauchy).
Divide the left hand side by $xyz=1$.
This leads to $(x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 9$, which is easy to show. Namely, $x+y+z\geq 3\sqrt[3]{xyz}$ by AM-GM and $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\geq 3/... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4419906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
${\log _ {x}}8 - {\log _{4x}} 8 = {\log _{2x}} 16$
${\log _ {x}}8 - {\log _{4x}} 8 = {\log _{2x}} 16$
I tried solving this problem by change of base and by $\frac{1}{\log{x}}$, but I really cannot seem to solve it no matter how hard I try.
I could only answer it by substituting $x$ for mcq answers.
Where do I even be... | HINT
As @TheoBendit has mentioned in the comments, I would recommend you to start with noticing
\begin{align*}
\log_{x}(8) - \log_{4x}(8) = \log_{2x}(16) & \Longleftrightarrow \frac{\log_{2}(8)}{\log_{2}(x)} - \frac{\log_{2}(8)}{\log_{2}(4x)} = \frac{\log_{2}(16)}{\log_{2}(2x)}
\end{align*}
where $x > 0$ and $x\not\in\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4421243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Maximize $z$ over $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$ Suppose that $x$, $y$, and $z$ are real numbers such that $x + y + z = 3$ and $x^2 + y^2 + z^2 = 6$. What is the largest possible value of $z$?
I tried applying Cauchy-Schwarz to get $(x^2+y^2+z^2)(1+1+1)\geq (x+y+z)^2$, but this doesn't say anything. I also ... | Define $r = [x,y,z]^T$, then your objective function using Lagrange multipliers is
$ f(r,\lambda_1, \lambda_2) = c^T r + \lambda_1 (b^T r - 3) + \lambda_2 (r^T r - 6) $
where $c = [0, 0, 1]^T , b = [1, 1, 1]^T $
Differentiating, and equating to zero:
$\nabla_r f = c + \lambda_1 b + 2 \lambda_2 r = 0 \hspace{15pt}(1) $
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4423472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
$\frac{1}{a+b}+\frac{1}{a+c}=\frac{3}{a+b+c}$ in a triangle Find the angle $\alpha$ of a triangle with sides $a,b$ and $c$ for which the equality $$\dfrac{1}{a+b}+\dfrac{1}{a+c}=\dfrac{3}{a+b+c}$$ holds.
My idea is to use the law of cosines: $$\cos\alpha=\dfrac{b^2+c^2-a^2}{2bc}$$ after simplifying the given equality a... | Multiplying both sides of the equation by $(a+b)(a+c)(a+b+c)$, we get \begin{align*}
(a+c)(a+b+c)+(a+b)(a+b+c)&=3(a+c)(a+b)\\
a^2 + a b + 2 a c + b c + c^2+a^2 + 2 a b + a c + b^2 + b c&=3 a^2 + 3 a b + 3 a c + 3 b c\\
2 a^2 + 3 a b + 3 a c + b^2 + 2 b c + c^2&=3 a^2 + 3 a b + 3 a c + 3 b c\\
b c &=-a^2+ b^2 + c^2\\
\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4427001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Evaluating $\lim_{x\to\infty}\frac{f^{-1}(x)}{\ln(x)}$, where $f(x)=e^x+x^3-x^2+x$
$$f(x)=e^x+x^3-x^2+x$$
What is the following?
$$\lim_{x\to\infty}\frac{f^{-1}(x)}{\ln(x)}$$
Do I need to calculate the inverse of $f(x)$ or is there some other way that this limit can be solved?
| From $f'(x) = e^x + 3 x^2 - 2 x+1$ we deduce that $f'(x)\ge 3x^2 -2 x + 1>0$, hence $f$ is a bijection from ${\mathbb R}$ onto $f({\mathbb R}) = {\mathbb R}$ and we have $f^{-1}(x)\to +\infty$ when $x\to +\infty$. Let $y = f^{-1}(x)$, we have
\begin{equation}
\frac{\ln x}{y} = \frac{\ln(f(y))}{y} = \frac{y + \ln(1 + e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4429053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Find the number of ways in which 4 letters can be selected from the 9 letters of the word ALGORITHM, given at least one vowel is included. My solution:
3 cases:
1 vowel:
$6P3 \times 3C1 = 360 $
2 vowels:
$6P2 \times 3C2 \times 2! = 180 $
3 vowels:
$6P1 \times 3! = 36 $
Total number of ways: 360 + 180 + 36 = 576
Am I co... | Using the Inclusion-Exclusion principle, we count the number of combinations that use $1$ vowel, subtract those that use $2$, and add those that use all $3$.
The generating function for this is
$$[x^4]:(1-(1-x)^3)\frac{1}{(1-x)^6}$$
where
$$1-(1-x)^3=\binom{3}{1}x-\binom{3}{1}x^2+x^3$$
$$\frac{1}{(1-x)^6}=\binom{5}{5}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4432572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to describe Image of matrix function f by eigenvectors given
$$
\textbf{A}=\begin{bmatrix}
4& 1& 2\\
2 & 2 & 4\\
1&1&2
\end{bmatrix}
$$
$f({\bf x})=A{\bf x}, {\bf x}\in R^3$
I know eigenvectors : $v_1=[0,-2,1]^T$, $v_2=[-2,2,1]^T$, $v_3=[2,2,1]^T$
eigenvalues: 0,2,6
the kernel is solution of Ax=0, $x=[0, -2t,t]^T,... | Given that
$$
A=\left[ \begin{array}{ccc}
4 & 1 & 4 \\
2 & 2 & 4 \\
1 & 1 & 2 \\
\end{array} \right]
$$
Note that $A$ has rank two.
(Obviously, the second row is twice the third row. In other words,
$R_2 = 2 R_3$ or that $0 R1 + R_2 - 2 R_3 = 0$. Hence, the rows of $A$
are linearly dependent)
A simple calculation gives... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4434491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find $x,y,z$ satisfying $x(y+z-x)=68-2x^2$, $y(z+x-y)=102-2y^2$, $z(x+y-z)=119-2z^2$ Solve for $x,y,z$:
$$x(y+z-x)=68-2x^2$$
$$y(z+x-y)=102-2y^2$$
$$z(x+y-z)=119-2z^2$$
After some manipulation, I obtain
$$xy+xz=68-x^2$$
$$yz+xy=102-y^2$$
$$xz+yz=119-z^2$$
After combining equations, I get
$$y=\frac{-51-x^2+z^2}{x-z}$$
T... | Your system is equivalent to
\begin{align}
x(y+z+x)&=68\\
y(z+x+y)&=102\\
z(x+y+z)&=119.
\end{align}
Therefore, in effect, you have
\begin{align}
xa&=68\\
ya&=102\\
za&=119\\
a&=x+y+z.
\end{align}
This means that
$$a=\frac{68}{a}+\frac{102}{a}+\frac{119}{a}$$
or $$a^2=68+102+119\implies a=\pm 17,$$
which results in eit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4434581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
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.