Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Determining if a Telescoping Series is convergent or not How do you determine if a telescoping series is convergent or not? If it converges, what value does it converge to?
It seems like you need to do partial fraction decomposition and then evaluate each term individually?
For example:
$$ \sum_{n=2}^\infty \frac{1}{n... | Partial fraction decomposition in your example yields
$$
\frac{1}{n^3 - n} = \frac{1}{2} \left[ \frac{1}{n - 1} - \frac{2}{n} + \frac{1}{n + 1} \right]
$$
To see the telescoping of the sum, it's nice to arrange terms like this:
$$
\begin{array}{*{13}{c}}
2 \sum_{n = 2}^\infty \frac{1}{n^3 - n} &=& \frac{1}{1} &-& \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/351627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Please tell me if i did this inverse laplace correctly. Thanks The question is : find the inverse laplace transformation of
$$\frac{13s^2+3s+6}{(s-2)(s^2+9)}.$$
Please tell me if i did this correctly
Here is my work:
Using partial factions:
\begin{align}
Y(s) &= \frac{13s^2 + 3s + 6}{(s - 2)(s^2 + 9)}
\\
&= \frac{64}{... | The partial fraction expansion yields:
$\displaystyle
\frac{3 (35 s+83)}{13 (s^2+9)} + \frac{64}{13 (s-2)} = \frac{3(35 s)}{13 (s^2+9)} + \frac{3(83)}{13 (s^2+9)}+ \frac{64}{13 (s-2)}$
Now, we put that result into the desired forms:
$\displaystyle \frac{3(35 s)}{13 (s^2+3^2)} + \frac{3(83)}{13 (s^2+3^2)}+ \frac{64}{13... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof that the sum of the cubes of any three consecutive positive integers is divisible by three. So this question has less to do about the proof itself and more to do about whether my chosen method of proof is evidence enough. It can actually be shown by the Principle of Mathematical Induction that the sum of the cube... | Your approach and deduction is absolutely fine
Just for the sake of completeness, here is an alternative prove
Known
$$(a+b+c)^3 = a^{3} + b^{3} + c^{3} + 3 a^{2} b + 3 a^{2} c + 3 a b^{2} + 3 a c^{2} + 3 b^{2} c + 3 b c^{2} + 6 a b c $$
$$ = a^{3} + b^{3} + c^{3} + 3f(a,b,c)$$
Let $a=k-1,b=k,c=k+1$,then, $a,b,c$ rep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/354384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 6
} |
Equation of the line passing through the origin and parallel to the planes $x+y+z=-1$ and $x-y+z=1$
Find a vector equation of the line that passes through the origin and is parallel to the planes $x+y+z=-1, x-y+z=1$
Is the answer $2x-2z=0$? I took the normals of the two planes which are $(1, 1, 1)$ and $(1, -1, 1)$ a... | Your analysis is correct.
Here is a different way to describe the line parallel to the intersection of the two planes. Just row reduce the corresponding homogeneous system of equations. (These planes are parallel to your given ones but passing through the origin.)
$$
\left\{
\begin{align}
x + y + z &= 0 \\
x - y + z ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/355520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving $\binom{2n}{n}\le 4^n$ for all $n$ by smallest counterexample
Prove $$\binom{2n}{n}\le 4^n$$ for all natural numbers $n$ by smallest (minimal) counterexample.
My attempt:
First, $$\binom{2n}n = \frac{(2n)!}{(n!)^2} \le 4^n\;.$$ We know that $x\ne 0$ because $\frac{(2\cdot 0)!}{(0!)^2} = 1$ which is true. So $... | HINT: In order to complete your proof, you need to show that the inequality $$\frac{(2x-2)!}{((x-1)!)^2} \le 4^{x-1}\tag{1}$$ implies that $$\frac{(2x)!}{x!^2}\le 4^x\;.$$
Now $$\frac{(2x)!}{x!^2}=\frac{2x(2x-1)(2x-2)!}{x^2(x-1)!^2}=\frac{2x(2x-1)}{x^2}\cdot\frac{(2x-2)!}{(x-1)!^2}\le\frac{2x(2x-1)}{x^2}\cdot4^{x-1}$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Really Confused on a surface area integral can't seem to finish the integral off. Basically the question asks to compute $\int \int_{S} ( x^{2}+y^{2}) dA$ where S is the portion of the sphere $x^{2} + y^{2}+ z^{2}= 4$ and $z \in [1,2]$ we start with a chnage of variables
$x=x $
$y=y$
$ z= 2 \cdot(4-(x^{2} + y^{2}... | Although the sine substitution probably is the easiest method, here's another one:
If the integrand is a fraction with a square root in the denominator, see if you can write the integrand as the derivative of the square root times another function to pave the way for an integration by parts:
$I = \int \frac{r^3}{\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/356385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Is $2^{218!} +1$ prime?
Prove that $2^{218!} +1$ is not a prime number.
I can prove that the last digit of this number is $7$, and that's all.
Thank you.
| Hint $\ $ If $\rm\: k\:$ is odd then $\rm\:a^n\!+\!1\mid a^{nk}\!+\!1\ $ by $\rm\ mod\ a^n\!+\!1\!:\ a^n\!\equiv -1\:\Rightarrow\:a^{nk}\!\equiv (a^n)^k\!\equiv (-1)^k\equiv -1.\:$
Or $ $ Factor Theorem $\rm\:\Rightarrow\: x\!-\!c\mid x^k\!-\!c^k\: $ in $\rm\:\Bbb Z[x],\:$ so $\rm\:c=-1\:\Rightarrow\: x\!+\!1\mid x^k\!... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/357037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 7,
"answer_id": 3
} |
Finding a point in an ellipsoid I know the semi-principal axes $(x,y,z)$ of the ellipsoid $E$ (centered at the origin). Given the normalized direction vector $\vec{v}=(a,b,c)$ pointing from the origin to the surface, how can I find the factor $f$ so that $f\vec{v} \in E$?
| The ellipsoid $E$ is the locus of points $(x, y, z) \in \mathbb{R}^3$ satisfying
$$
\frac{x^2}{r^2} + \frac{y^2}{s^2} + \frac{z^2}{t^2} = 1
$$
If $\vec{v} = (a, b, c)$ and $f\vec{v} \in E$ with $f \in \mathbb{R}$, then we can find the scalar $f$ by putting the coordinates of $f\vec{v}$ into the equation.
$$
f\vec{v} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/358697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Minimum Value of expression Given that $x$, $y$ and $z$ are positive real numbers satisfying $xyz=32$, find the minimum value of:
$$x^2+4xy+4y^2+2z^2$$
Perhaps AM-GM and manipulation but I'm not quite sure how?
Source BMO.
| Yes AM-GM is the right approach.
By AM-GM,
$$\frac{x^2+2xy+2xy+4y^2+z^2+z^2}{6} \geq (16 \cdot x^4 \cdot y^4 \cdot z^4)^{\frac{1}{6}}=$$
$$=(2^{24})^{\frac{1}{6}}=16$$ with equality iff $x=z=2y$.
So the minimum value of $x^2+4xy+4y^2+2z^2$ is $16 \cdot 6=96$. QED
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/359157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Finding the solution to this specific recurrence relation What would be the solution to $a_n = 7a_{n−2} + 6a_{n−3}$ with $a_0 = 9$,
$a_1 = 10$, and $a_2 = 32$
I can find it for a specific value of (n), but not for just a general solution. Thanks!
| Use generating functions. Define $A(z) = \sum_{n \ge 0} a_n z^n$, and write the recurrence as:
$$
a_{n + 3} = 7 a_{n + 1} + 6 a_n \quad a_0 = 9, a_1 = 10, a_2 = 32
$$
By properties of ordinary generating funtions:
$$
\frac{A(z) - a_0 - a_1 z - a_2 z^2}{z^3} = \frac{A(z) - a_0}{z} + 6 A(z)
$$
Writing as partial fraction... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/362522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find $(a, b)$ such that $\lim_{x \to 0} \frac{ax -1 + e^{bx}}{x^2} = 1$ Find $(a, b)$ such that $\lim_{x \to 0} \frac{ax -1 + e^{bx}}{x^2} = 1$
I am able to find $b = \pm \sqrt{2}$ using L'Hopitals Rule, but unable to do anything for $a$.
| Another way to do it is with Taylor expansion of the numerator,
$$-1 + ax + (1 + bx + \frac{b^2}{2}x^2 + \mbox{higher order terms}) = (a+b)x + \frac{b^2}{2} + \mbox{ h.o.t.}).$$
Now dividing by $x^2$ and sending $x \to 0$ it is clear that the higher order terms vanish, and $a = -b$ to make sure the limit exists. We are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Evaluating this integral $ \small\int \frac {x^2 dx} {(x\sin x+\cos x)^2} $ The question:
Compute$$
\int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2}
$$
Tried integration by parts. That didn't work.
How do I proceed?
| $$\text{Observe that, }\frac{d(x\sin x+\cos x)}{dx}=x\cos x$$
$$ \int \frac {x^2 \, \operatorname{d}\!x} {(x\sin x+\cos x)^2} =\int \frac x{\cos x}\cdot \frac{x\cos x}{(x\sin x+\cos x)^2}dx$$
So, if $z=x\sin x+\cos x, dz=x\cos xdx$
So, $\int \frac{x\cos x}{(x\sin x+\sin x)^2}dx=\int \frac{dz}{z^2}=-\frac1z=-\frac1{x\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 1
} |
Evaluating Sums Algebraically or Combinatorially Consider
(1) $$\sum_{k=0}^{n}\binom{n}{k}2^{k-n}$$
(2) $$\sum_{k=0}^{n}\binom{n}{k}\frac{k!}{(n+k+1)!}$$
These sums appear too difficult (in my mind) to evaluate combinatorially. What are some good methods to attack these problems algebraically?
| A combinatorial argument for (1) isn’t too hard. First rewrite the sum:
$$\sum_k\binom{n}k2^{k-n}=2^{-n}\sum_k\binom{n}k2^k\;.$$
Now $\binom{n}k2^k$ is the number of ways of choosing a $k$-subset $S$ of $[n]$ and then a subset $T$ of $S$, so $\sum_k\binom{n}k2^k$ is the number of ways of splitting $[n]$ into three subs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/366752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Determine the limiting behaviour of $\lim_{x \to \infty}{\frac{\sqrt{x^4+1}}{\sqrt[3]{x^6+1}}}$ Determine the limiting behaviour of $\lim_{x \to \infty}{\dfrac{\sqrt{x^4+1}}{\sqrt[3]{x^6+1}}}$
Used L'Hopitals to get $\;\dfrac{(x^6+1)^{\frac{2}{3}}}{x^2 \sqrt{x^4+1}}$ but not sure what more i can do after that.
| $$lim_{x \to \infty} \frac{\sqrt{x^4 + 1}}{\sqrt[3]{x^6+1}}$$
Mutliply top and bottom by $\frac{1}{x^2}$
$$lim_{x \to \infty} \frac{\sqrt{x^4 + 1}}{\sqrt[3]{x^6+1}} \cdot \frac{\frac{1}{x^2}}{\frac{1}{x^2}} = lim_{x \to \infty} \frac{\sqrt{\frac{x^4}{x^4} + \frac{1}{x^4}}}{\sqrt[3]{\frac{x^6}{x^6}+\frac{1}{x^6}}}$$
$$=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Why does the harmonic series diverge but the p-harmonic series converge I am struggling understanding intuitively why the harmonic series diverges but the p-harmonic series converges. I know there are methods and applications to prove convergence, but I am only having trouble understanding intuitively why it is. I know... | We produce two series that are close in spirit to the series you mentioned. Perhaps the divergence of the first, and the convergence of the second, will be clearer.
Consider the series
$$\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{16}+\frac{1}{16}+\frac{1}{16}+\frac{1}{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367135",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 7,
"answer_id": 5
} |
How to find $\cos A \cos B - \sin A \sin B$? Given that:
$\tan A=1$
and
$\tan B = \sqrt{3}$
How would you find $\cos A \cos B - \sin A \sin B$?
EDIT: This is what I've tried after reading bhattacharjee's answer:
$$ \tan(A+B) = \tan A+\tan B−\tan A\tan B$$
so,
$\tan(A+B)= {1+\sqrt{3} \over 1-\sqrt{3}}$
from this I g... | since $tan A =1$, then $A = 45^0$ and since $tan B = \sqrt{3}$, then $ B = 60^0$.
By drawing triangles you can find that $sin 45^0 = 1/\sqrt{2}$ and $cos 45^0 = 1/\sqrt{2}$ and $sin 60^0 = \sqrt{3}/2$ and $cos 60^0 = 1/2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/367477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Finding the complex trig integral using the method of residues for $\int_{-\pi}^{\pi}\frac{d\theta}{1 + (\sin^{2}\theta)} = {\pi}{\sqrt{2}}$ $$\int_{-\pi}^{\pi}\frac{d\theta}{1 + (\sin^{2}\theta)} = {\pi}{\sqrt{2}}$$
I can't seem to factor this question into the solution the textbook got
so $(\sin^{2}\theta)$ = ${((1/... | $$\dfrac1{1+\sin^2(\theta)} = \sum_{k=0}^{\infty}(-1)^k \sin^{2k}(\theta)$$
Now recall that
$$\int_{-\pi}^{\pi} \sin^{2k}(\theta) d \theta = \dfrac{(2k)!}{(k!)^2 \cdot 4^k} 2 \pi$$
Hence,
$$I = \int_{-\pi}^{\pi} \dfrac{d \theta}{1+\sin^2(\theta)} = \underbrace{\int_{-\pi}^{\pi}\sum_{k=0}^{\infty}(-1)^k \sin^{2k}(\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/367709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Derivation of the general forms of partial fractions I'm learning about partial fractions, and I've been told of 3 types or "forms" that they can take
(1) If the denominator of the fraction has linear factors:
$${5 \over {(x - 2)(x + 3)}} \equiv {A \over {x - 2}} + {B \over {x + 3}}$$
(2) If the denominator of the frac... | To look at your third problem, consider the following example: $${{x^2 + 4} \over {(x)(x + 2){{(3x-2)}}}} = {A \over {x}} + {B \over {x + 2}} + {C \over {{{(3x-2)}}}}$$
Next, set the numerators equal. $$x^2+4 = A(x+2)(3x-2) + Bx(3x-2) + Cx(x+2)$$ Now pick few x values and find the constants. For $x=0,$ $4=A(2)(-2).$ So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/368665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 8,
"answer_id": 5
} |
Find expansion around $x_0=0$ into power series and find a radius of convergence My task is as in the topic, I've given function $$f(x)=\frac{1}{1+x+x^2+x^3}$$
My solution is following (when $|x|<1$):$$\frac{1}{1+x+x^2+x^3}=\frac{1}{(x+1)+(x^2+1)}=\frac{1}{1-(-x)}\cdot\frac{1}{1-(-x^2)}=$$$$=\sum_{k=0}^{\infty}(-x)^k\c... | Hint: Another method to approach the problem (more by brute force than Andre's clever technique) is $$f(x)=\frac{1}{1+x+x^2+x^3}=\frac{1}{(x+1)(x^2+1)}.$$ Now use partial fraction decomposition and geometric series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/369435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
What's the Maclaurin Series of $f(x)=\frac{1}{(1-x)^2}$? This function seemed to be pretty much straight forward, but my solution is incorrect.
I have two questions:
1. Where did I make a mistake?
2. I learned that there are shortcuts for finding a series (substitution / multiplication / division / differentiation / in... | Recall that
$$f(x) = f(0) + \dfrac{f'(0)}{1!} x + \dfrac{f''(0)}{2!}x^2 + \cdots$$
Your derivative computation is correct, i.e., $f^{(n)}(0) = (n+1)!$. But you have them matched to the wrong denominators. Hence, you will get
$$\dfrac1{(1-x)^2} = 1 + \dfrac{2!}{1!}x + \dfrac{3!}{2!}x^2 + \dfrac{4!}{3!}x^3 + \cdots = 1 +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/369889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Finite difference implicit schema for wave equation 1D not unconditionally stable? The wave equation 1D with constant density is defined as:
\begin{equation}
\frac{\partial^2 U}{\partial t ^2} = V^2 \frac{\partial^2 U}{\partial x ^2}
\label{eqa}
\end{equation}
And the implicit difference schema:
\begin{equation}
U_j^{n... | Your method for Neumann stability analysis is correct, and indeed I would expect the scheme to be unconditionally stable.
I do not understand the step where you introduce the sine. I think that
$$ e^{ix} - 2 + e^{-ix} = -4\sin^2(\tfrac12x). $$
The minus sign seems to have been lost in your computation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/372969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to derive the Golden mean by using properties of Gamma function? The Golden mean known as $\frac{1+\sqrt{5}}{2}$.
How could one show the Golden mean can be expressed as
$$
\frac{2\cdot 3\cdot 7\cdot 8\cdot 12\cdot 13\cdots}{1\cdot 4\cdot 6\cdot 9\cdot 11\cdot 14\cdots}
$$
This is the limiting case of the Rogers-Ram... | Consider the combination
\begin{align}f_N=\frac{\Gamma\left(N+\frac{2}{5}\right)\Gamma\left(N+\frac{3}{5}\right)}{\Gamma\left(N+\frac{1}{5}\right)\Gamma\left(N+\frac{4}{5}\right)}=
\frac{\left(5N-2\right)\cdot\left(5N-3\right)}{\left(5N-1\right)\cdot\left(5N-4\right)}f_{N-1}=\ldots=\\=
\frac{\left(5N-2\right)\cdot\left... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable? In this recent answer to this question by Eesu, Vladimir
Reshetnikov proved that
$$
\begin{equation}
\left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1}
\end{equation}
$$
I would like to know if this resu... | This kind of simplification occurs if and only if $p \pm q\sqrt d$ has a cube root of the form $x \pm y\sqrt d$ with rational $x,y$. So, to get all instances of this, start by choosing $x+y\sqrt d$, and cube it to get the values for $p$ and $q$.
Setting up the system $(x + y\sqrt d)^3 = p + q\sqrt d$, we get $x^3+3dxy^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/374619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
"answer_count": 6,
"answer_id": 0
} |
Generating functions of partition numbers I don't understand at all why:
\begin{equation}
\sum\limits_{n=0}^\infty p_n x^n = \prod\limits_{k=1}^\infty (1-x^k)^{-1}
\end{equation}
Where $p_n$ is the number of partitions of $n$. Specifically how can the sum of positive numbers ever be factored into a fraction or factors ... | We have $\frac{1}{1-x^k}=1+x^k+x^{2k}+x^{3k}+\cdots$. These are combined formally in a process known as generating functions.
More details, as requested. We multiply out $(1+x+x^2+x^3+\cdots)(1+x^2+x^4+x^6+\cdots)(1+x^3+x^6+x^9+\cdots)(1+x^4+x^8+\cdots)\cdots$, and gather the powers of $x$, as if they were polynomial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/375863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\int_0^{\infty} \text{sinc}^m(x) dx$ How do I evaluate $$I_m = \displaystyle \int_0^{\infty} \text{sinc}^m(x) dx,$$ where $m \in \mathbb{Z}^+$?
For $m=1$ and $m=2$, we have the well-known result that this equals $\dfrac{\pi}2$. In general, WolframAlpha suggests that is seems to be a rational multiple of $\p... | Notice $\lim_{x\to 0} \frac{\sin x}{x}$ is bounded at $x = 0$,
$$\begin{align}\int_0^{\infty} \left(\frac{\sin x}{x}\right)^m dx
&= \frac12 \int_{-\infty}^{\infty} \left(\frac{\sin x}{x}\right)^m dx\tag{*1}\\
&= \lim_{\epsilon\to 0} \frac12 \left(\frac{1}{2i}\right)^m \oint_{C_{\epsilon}} \left(\frac{e^{ix} - e^{-ix}}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 1
} |
Evaluate $ \lim\limits_{x \to 0} \frac{1}{\sin^3x}\int_0^x{\sin(t^2) } dt$
$$ \lim_{x \to 0} \frac{1}{\sin^3x}\int_0^x{\sin(t^2) dt}$$
This is what I've tried:
Let $F(x) = \displaystyle\int_0^x{\sin(t^2) dt}$, and let $f(x) = {\sin(t^2)}$.
Then $F'(x) = f(x) \Rightarrow F'(0) = f(0) = \sin 0 = 0$.
Where do I go f... | Using L'Hospital's rule once, we have
$$
\lim_{x\rightarrow 0}\frac{\sin x^2}{3\sin^2x \cos x}.
$$
We examine the limit
$$
\lim_{x\rightarrow 0} \frac{\sin x^2}{\sin^2x} = \lim_{x\rightarrow 0} \frac{2x\cos x^2}{2\sin x \cos x} =\lim_{x\rightarrow 0} \frac{x\cos x}{\sin x} = 1
$$
using the fact that $\lim_{x\rightarrow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/379508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Calculus Reduction Formula
For any integer $k > 0$, show the reduction formula
$$\int^{2}_{-2} x^{2k} \sqrt{4-x^2} \, dx = C_k \int^{2}_{-2} x^{2k-2} \sqrt{4-x^2} \, dx$$
for some constant $C_{k}$.
(original image)
I thought this would be fairly straightforward but im a little confused. Do I start out by doing a ... | I like to find the reduction formula for the integral by rationalization followed by integration by parts.
$$
\begin{aligned}
I_{k} &=\int_{-2}^{2} x^{2 k} \sqrt{4-x^{2}} d x \\
&=\int_{-2}^{2} \frac{x^{2 k}\left(4-x^{2}\right)}{\sqrt{4-x^{2}}} d x \\
&=-\int_{-2}^{2} x^{2 k-1}\left(4-x^{2}\right) d\left(\sqrt{4-x^{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/380025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to find the number of real roots of the given equation?
The number of real roots of the equation $$2 \cos \left( \frac{x^2+x}{6} \right)=2^x+2^{-x}$$ is
(A) $0$, (B) $1$, (C) $2$, (D) infinitely many.
Trial: $$\begin{align} 2 \cos \left( \frac{x^2+x}{6} \right)&=2^x+2^{-x} \\ \implies \frac{x^2+x}{6}&=\cos ^{-... | *
*$RHS=2^x+2^{-x}=2^x+\frac{1}{2^x}\geq 2$ by $AM\geq GM$, since $2^x>0$ for all $x\in\mathbb{R}$
*$LHS\leq 2$ for all $x\in\mathbb{R}$. This is obvious.
*One can verify $x=0$ is a solution.
*$f(x)=2^x+2^{-x}$ implies $f'(x)>0$ for $x>0$.
These should suffice to conclude the unique solution at $x=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/380896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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Prove that if $a, b, c$ are positive odd integers, then $b^2 - 4ac$ cannot be a perfect square. Prove that if $a, b, c$ are positive odd integers, then $b^2 - 4ac$ cannot be a perfect square.
What I have done:
This has to either be done with contradiction or contraposition, I was thinking contradiction more likely.
| $$a,b,c=\text{odd}\quad\iff\quad a=2A+1\quad;\quad b=2B+1\quad;\quad c=2C+1$$ $$\Delta=b^2-4ac=n^2\quad\iff\quad(2B+1)^2-4\,(2A+1)(2C+1)=n^2$$ $$(4B^2+4B+1)-4\,(4AC+2A+2C+1)=n^2$$ $$4\,\Big[(B^2+B)-(4AC+2A+2C+1)\Big]+1=n^2$$ $$\iff n=2k+1\iff n^2=4k^2+4k+1\iff$$ $$\underbrace{\underbrace{B(B+1)}_{even}-\underbrace{2\,(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/382564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 2
} |
Inverse of a symmetric tridiagonal filter matrix How to get the inverse of this matrix:
$\left(\begin{array}{ccccccc}
2&-1\\-1&2&-1\\&-1&2&-1\\&&&\ddots\\&&&&\ddots\\&&&&-1&2&-1\\&&&&&-1&2
\end{array}\right)$
where the blank elements are all zeros. Thank you.
| Lets do an example using your matrix with dimension $4x4$, symmetric, tridiagonal.
We have the matrix:
$$\displaystyle A = \begin{bmatrix}2&-1&0&0\\-1&2&-1&0\\0&-1&2&-1\\0&0&-1&2\end{bmatrix}.$$
The inverse of this matrix:
$$\displaystyle A^{-1} = \frac{1}{5}\begin{bmatrix}4 & 3 & 2 & 1\\3 & 6 & 4 & 2\\2 & 4 & 6 & 3\\1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Integral Of $\int\sqrt{\frac{x}{x+1}}dx$ I want to solve this integral
$$\int\sqrt{\frac{x}{x+1}}dx$$
And think about:
1) $t=\frac{x}{x+1}$
2) $dt = (\frac{1}{x+1} - \frac{x}{(x+1)^2})dx$
Now I need your advice! Thanks!
| Let $$\sqrt{\dfrac{x}{x+1}} = t \implies \dfrac{x}{x+1} = t^2 \implies x+1 = \dfrac1{1-t^2} \implies dx = \dfrac{2tdt}{(t^2-1)^2}$$
Hence,
$$\int \sqrt{\dfrac{x}{x+1}} dx = \int \dfrac{2t^2}{(t^2-1)^2}dt$$
I trust you can take it from here using partial fractions.
$$\dfrac{2t^2}{(t^2-1)^2} = \dfrac12 \left(\dfrac1{(t-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/385274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
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What is the most elementary proof that $\lim_{n \to \infty} (1+1/n)^n$ exists? Here is my candidate for
the most elementary proof that $\lim_{n \to \infty}(1+1/n)^n $ exists.
I would be interested in seeing others.
$***$ Added after some comments:
I prove here by very elementary means that the limit exists.
Calling th... | We can prove directly from BI at the expense of some algebra.
$(1+(v/u-1)/(n+1))^{n+1} > v/u$ with $u=1+1/n$ and $v=1$ is
$$\frac{1}{1+\frac{1}{n}} < \left(1+\frac{\frac{1}{1+\frac{1}{n}}-1}{n+1}\right)^{n+1} = \left( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right)^{n+1}$$
which gives $a_{n+1} >a_n$ in Marty's notation. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/389793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
} |
Solving recurrence relation, $a_n=6a_{n-1} - 5a_{n-2} + 1$ I'm trying to solve this recurrence relation:
$$
a_n = \begin{cases}
0 & \mbox{for } n = 0 \\
5 & \mbox{for } n = 1 \\
6a_{n-1} - 5a_{n-2} + 1 & \mbox{for } n > 1
\end{cases}
$$
I calculated generator function as:
$$
A = \frac{31x - 24x^2}{1 - ... | Let's write your recurrence relation for $n$ and $n+1$:
$a_{n}-6a_{n-1}+5a_{n-2}-1=0$
$a_{n+1}-6a_{n }+5a_{n-1}-1=0$
Now we subtract one from another:
$a_{n+1}-7a_{n }+11a_{n-1}-5a_{n-2}=0$ (relation 2)
Then, from Theorem on wiki we build a characteristic polynomial
$x^3-7x^2+11x-5 $ with roots $1,1,5$. Hence, by that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/390644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Solve $(x^2 + 5)^2 - 15(x^2 + 5) + 54 = 0$ I got the square root of 14 and 11 but the answer book states that these answers are wrong. Can someone help me? I used this formula to find the individual roots
$x = -\frac{p}{2} \pm \sqrt{(\frac{p}{2})^2 - q}$
| Using middle term factor, $$(x^2+5)^2-(9+6)(x^2+5)+6\cdot9=0$$
$$\implies (x^2+5)(x^2+5-9)-6(x^2+5-9)=0$$
$$\implies (x^2-4)(x^2-1)=0$$
$\implies x^2-1=0$ or $x^2-4=0$
Alternatively, using quadratic equation formula for $x^2+5=\frac{15\pm\sqrt{15^2-4\cdot1\cdot 54}}{2\cdot1}=\frac{15\pm 3}2=9$ or $6$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/392324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Evaluation of a specific determinant.
Evaluate $\det{A}$, where $A$ is the $n \times n$ matrix defined by $a_{ij} = \min\{i, j\}$, for all $i,j\in \{1, \ldots, n\}$.
$$A_2
\begin{pmatrix} 1& 1\\
1& 2
\end{pmatrix};
A_3 = \begin{pmatrix} 1& 1& 1\\
1& 2& 2\\
1& 2& 3
\end{pmatrix};
A_4 = \begin{pmatrix} ... | $$A_2 = \begin{bmatrix}1 & 1\\ 1 & 2 \end{bmatrix} = \begin{bmatrix}1 & 0\\ 1 & 1\end{bmatrix}\begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}$$
$$A_3 = \begin{bmatrix} 1 & 1 & 1\\ 1 & 2 & 2\\1 & 2 & 3 \end{bmatrix} = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1\end{bmatrix}\begin{bmatrix}1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/392738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Find $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ if $a+b+c=0$ I'm stuck at this algebra problem, it seems to me that's what's provided doesn't even at all.
Provided: $$a+b+c=0$$
Find the value of: $$\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$$
Like I'm not sure where to start, and t... | $$ −a^2b−b^2c−c^2a+a^2c+b^2a+c^2b
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\\
\overset{\tiny{+abc-abc}}{=}
\Big[ abc+bc^2-b^2c-ac^2 \Big]
+
\Big[ -a^2b-abc+ab^2+a^2c \Big]
\\
=c(ab+bc-b^2-ac)-a(ab+bc-b^2-ac)
\\
=(c-a)(ab+bc-b^2-ac)
\\
=(c-a)(a-b)(b-c)
. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/393786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Details about a Recurrence Relation problem. I am trying to understand Recurrence Relations through the Towers of Hanoi example, and I am having trouble understanding the last step:
If $H_n$ is the number of moves it takes for n rings to be moved from the first peg to the second peg, then:
$H_n = 2H_{n−1} + 1$
$H_n = 2... | The last few lines contain errors. The line
$$H_n=2^{n-1}H_1+2^{n-2}+2^{n-3}+\ldots+2+1$$
is correct, but the next line is not: since $H_1=1$, it should be
$$H_n=2^{n-1}+2^{n-2}+2^{n-3}+\ldots+2+1\;.$$
Apparently the exponents got dropped down onto the main line of text. This is now the sum of a geometric series:
$$H_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/394757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Limit $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}$ I want to compute $$\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$$
Is it OK how I want to do?
$$\sin{\sqrt{x+1}}-\sin{\sqrt{x}}=2\sin{\frac{\sqrt{x+1}-\sqrt{x}}{2}}\cos{\frac{\sqrt{x+1}+\sqrt{x}}{2}}=2\sin{\frac{1}{2(\sqrt{x+1}+\sqrt{x})}}\cos{\fra... | Here's another approach, using the OP's trick of getting the square roots into the denominator, but with a different trig identity, in this case $\sin(a+b)=\sin a\cos b+\cos a\sin b$.
$$\begin{align}
|\sin(\sqrt{x+1})-\sin\sqrt x|
&=|\sin(\sqrt x+\sqrt{x+1}-\sqrt x)-\sin\sqrt x|\\
&=|\sin\sqrt x ( \cos(\sqrt{x+1}-\sqrt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/394899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 2
} |
quadratic equation precalculus from Stewart, Precalculus, 5th, p56, Q. 79
Find all real solutions of the equation
$$\dfrac{x+5}{x-2}=\dfrac{5}{x+2}+\dfrac{28}{x^2-4}$$
my solution
$$\dfrac{x+5}{x-2}=\dfrac{5}{x+2}+\dfrac{28}{(x+2)(x-2)}$$
$$(x+2)(x+5)=5(x-2)+28$$
$$x^2+2x-8=0$$
$$\dfrac{-2\pm\sqrt{4+32}}{2}$$
$$\dfrac{... | Why do we have to reject the solution $x=2$?
Hint: What happens when we put $x=2$ in the original equation?
Review your equations. Make sure that you didn't multiply by 0.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/395691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How to evaluate $\sqrt[3]{a + ib} + \sqrt[3]{a - ib}$? The answer to a question I asked earlier today hinged on the fact that
$$\sqrt[3]{35 + 18i\sqrt{3}} + \sqrt[3]{35 - 18i\sqrt{3}} = 7$$
How does one evaluate such expressions? And, is there a way to evaluate the general expression
$$\sqrt[3]{a + ib} + \sqrt[3]{a - ... | In polar form,
$$x = 35+18 i \sqrt{3} = \sqrt{13}^3 \cdot \exp\left( i \cdot \rm{atan} \frac{18\sqrt{3}}{35}\right)$$
so that
$$\sqrt[3]{x} + \sqrt[3]{\overline{x}} = 2 \sqrt{13} \; \cos\left(\frac{1}{3} \rm{atan} \frac{18\sqrt{3}}{35} \right)$$
which magically evaluates to $7$ in a calculator.
(I know, I know: It's n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/396915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 6
} |
Finding the range of a vector valued function For a single valued function, I can infer if the function is monotone from its derivative.
For a vector valued function, is it possible to infer monotonicity from the directional derivative?
For example, define
$$
D=[1,2]\times[1,2],
$$
and
$$
f(x,y)=\left( \frac{2}{1/x+1... | Note that the Jacobian determinant is
$$\det J_f(x,\, y)=\det\begin{bmatrix}
\frac{2}{\left(1+x/y\right)^2} & \frac{2}{\left(1+y/x\right)^2} \\
\frac{y^{1/2}}{2x^{1/2}} & \frac{y^{1/2}}{2x^{1/2}}
\end{bmatrix}=\\
=\dfrac{1}{{\left(\dfrac{1}{x} + \dfrac{1}{y}\right)}^{2} \sqrt{x y} x} - \dfrac{1}{{\left(\dfrac{1}{x} + \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/397261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Value of series, Partialsum? given is the following series
$$\sum_{n=1}^\infty \frac{2n+1}{n^2(n+1)^2}$$
And I need to find its value.
How can I start finding it?
Thanks for all
does the Telescop-Summing work here as well?:
$\sum_{n=1}^\infty \frac{1}{4n^2-1} $
now:
$\frac{1}{4n^2-1} = \frac{1}{2} * \frac{(2n+1)-(2n-1... | split it up the numerator to 2n + 2 - 1 then factor to 2(n+1) -1.
By spliting the fraction you can cancel terms.
This should allow you to see this is a finite result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/398075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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How to resolve this algebra equation? $$f = X^3 - 12X + 8$$ $a $- complex number, $a$ is a root for $f$
$b = a^2/2 - 4 $.
Show that $f(b) = 0$
This is one of my theme exercises ... Some explanations will be appreciated ! Thank you all for your time .
| \begin{align*}
f(b) &= f(\frac{a^2}{2} - 4) \\
&= (\frac{a^2}{2} - 4)^3 - 12(\frac{a^2}{2} - 4) + 8 \\
&= \frac{a^6}{8} - 3a^4 + 24 a^2 - 64 - 6a^2 + 48 + 8 \\
&= \frac{a^6}{8} -3a^4 + 18a^2 -8
\end{align*}
On the other hand,
$$
f(a) = a^3 - 12a + 8 = 0
$$
and therefore
\begin{align*}
0 = 0^2 &= (a^3 - 12a + 8)^2 \\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/399064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Linear equation of 4 variables I'm stuck on this Math problem :
How many solutions does the equation
$x_{1} + x_{2} + 3x_{3} + x_{4} = k$
have, where $k$ and the $x_{i}$ are non-negative integers such that $x_{1} \geq 1$, $x_{2} \leq 2$, $x_{3} \leq 1$
and $x_{4}$ is a multiple of 6.
I tried to write the possible c... | This will give you a start: it's the coefficient of $x^k$ in $$(x+x^2+x^3+\dots)(1+x+x^2)(1+x^3)(1+x^6+x^{12}+\dots)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/399204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proving the remainder is $1$ if the square of a prime is divided by $12$ Given, $p$ is a prime number and $p>3$. How do we prove that the remainder $r$ is always $1$ if $p^2$ is divided by $12$?
| If $(a,12)=1, (a,3)=1$ and $ (a,2)=1$
$(a,3)=1\implies a\equiv\pm1\pmod 3\implies a^2\equiv1\pmod 3$
$(a,2)=1\implies a$ is odd $=2b+1$(say) where $b$ is some integer
$(2b+1)^2=4b^2+4b+1=8\frac{b(b+1)}2+1\equiv1\pmod 8$
$\implies a^2\equiv1\pmod { \text{lcm}(3,8)}$
Now, lcm $(3,8)=24$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/400391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
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Proving $\prod _{k=j}^n \frac{p_{k+1}}{p_k} = \frac{p_{n+1}}{p_j}\!\!,\;\;1\le j\!<\!n$ Let $p_n$ denote the $n$th prime number.
How could one prove that:
$$\prod \limits_ {k=j}^n \frac{p_{k+1}}{p_k} = \frac{p_{n+1}}{p_j}\!\!,\;\;1\le j\!<\!n$$
Examples:
$n=3812,\;j=81\qquad\implies\quad\large{\prod \limits _{k=81}^{3... | $$\prod _{k=j}^n \frac{p_{k+1}}{p_k} = \frac{p_{n+1}}{p_n} \times \frac{p_{n}}{p_{n-1}} \dots \frac{p_{j+2}}{p_{j+1}}\times \frac{p_{j+1}}{p_j}=\frac{p_{n+1}}{p_j}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/403577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Proof of Astroid? How can I prove that an astroid is an envelope of all line segments of length 1 from the x-axis to the y-axis?
I read one proof of this online at the link Link
but I don't understand how this proof works.
Therefore, it would be much appreciated if someone could show me another proof or make the online... | For a reverse proof, compute the tangent of the astroid, find the intersection with the axis and show they distance is 1:
Start with the parametric equation $(x,y)=(\cos^3\theta, \sin^3\theta)$, whose derivative relative to $\theta$ is $(x',y')=(3 \cos^2\theta \sin\theta, -3 \sin^3\theta \cos\theta)$. The tangent at $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Finding the correct statements about $(5+2\sqrt{6})^{2n+1} = S + t$ with $S$ integer and $0 \leq t < 1$ Problem:
If $n$ is a positive integer and $(5+2\sqrt{6})^{2n+1} = S + t$, where $S$ is an integer and $0 \leq t < 1$, then
(a) $S$ is an odd integer
(b) $S + 1$ is not divisible by $9$
(c) The integer next above $(5... | Hint: note that $(5+2\sqrt 6)^{2n+1}+(5-2\sqrt 6)^{2n+1}$ is an integer, and $(5-2 \sqrt 6) \lt 1$, so a power of it will be small and positive.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Not so easy optimization of variables? What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to $2x^2+5xy+3y^2=2$ and $6x^2+8xy+4y^2=3$. (calculus is not allowed). I tried everything I could but whenever I got for example $or$ $x^2+y^2=f(y)$ or $f(x)$ the function $f$ would always be a concave up parabola... | Since this question popped up to the top of the stack recently, I thought I'd add another answer, since there is a way to solve the problem without needing to determine the intersection points themselves (although they can be obtained with a small amount of additional calculation).
The hyperbola is $ \ 2x^2 + 5xy + 3y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/407244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
How to find the following indefinite integral? $$ \int {dx \over {\sin^3 x+\cos^3 x}}$$
Can this integral be found by substitution or any other method such as complex number?
| $$I=\int \frac{dx}{\sin^3x+\cos^3x}=\int \frac{dx} {\left(\sin(x)+\cos(x)\right)\left(1-\sin(x)\cos(x)\right)}$$ Write $$\sin(x)\cos(x)= \frac{\left(1-(\sin(x)-\cos(x))^2\right)}{2}$$ So
$$ I=\int \frac{dx} {\left(\sin(x)+\cos(x)\right) \left(1-\frac{\left(1-(\sin(x)-\cos(x))^2\right)}{2}\right)}$$ Now Use Substitution... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/410330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Limit with roots I have to evaluate the following limit:
$$ \lim_{x\to 1}\dfrac{\sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2+1}-\sqrt{x^4+1}} . $$
I rationalized both the numerator and the denominator two times, and still got nowhere. Also I tried change of variable and it didn't work.
Any help is gratefu... | I will compute the limits
of the numerator and denominator separately.
To make this more rigorous, imagine that
the two limits are being done
at the same time,
so the final division is justified.
$\sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}
=\sqrt{x+1}(1+\sqrt{x-1}-\sqrt{x^2-x+1})
$
so,
putting $x = 1+y$
and using $\sqrt{1+z}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/411676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
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There are no integers $x,y$ such that $x^2-6y^2=7$ How to show that there is no here are no integers $x,y$ such that $x^2-6y^2=7$?
Help me. I'm clueless.
| We work modulo $8$. Clearly $x$ cannot be even. So $x^2\equiv 1\pmod{8}$.
If $y$ is even, then $6y^2\equiv 0\pmod{8}$, so $x^2-6y^2\equiv 1\pmod{8}$. But $7\not\equiv 1\pmod{8}$.
If $y$ is odd, then $y^2\equiv 1\pmod{8}$, so $6y^2\equiv 6\pmod{8}$. It follows that $x^2-6y^2\equiv -5\pmod{8}$. But $7\not\equiv -5\pmod{8... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/414870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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How to simplify $\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$? $$\frac{1-\frac{1-x}{1-2x}}{1-2\frac{1-x}{1-2x}}$$
I have been staring at it for ages and know that it simplifies to $x$, but have been unable to make any significant progress.
I have tried doing $(\frac{1-x}{1-2x})(\frac{1+2x}{1+2x})$ but that doesn't h... |
$$\frac{1 - \frac{1-x}{1 - 2x}}{1 - 2\frac{1-x}{1-2x}}$$
Multiplying through by $1-2x$ gives
$$\frac{1 - 2x - (1 - x)}{1 - 2x - 2(1 - x)}.$$
Expanding the brackets and then simplifying gives
$$\frac{1 - 2x - 1 + x}{1 - 2x - 2 + 2x}$$
$$= \frac{-x}{-1} = x.$$
NOTE: This method works well because you have the same "den... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
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How to find the integral of $\frac{1}{2}\int^\pi_0\sin^6\alpha \,d\alpha$ $$\frac{1}{2}\int^\pi_0\sin^6\alpha \,d\alpha$$
What is the method to find an integral like this?
| Let $x=\cos A+i\sin A$
Then we have,
$\displaystyle x-\frac{1}{x}=2i\sin A$
$\displaystyle (x-\frac{1}{x})^{6}=-2^6\sin^6 A$
Then we have by expanding,
$x^6+\frac{1}{x^6}-6(x^4+\frac{1}{x^4})+15(x^2+\frac{1}{x^2})-20=2\cos6 A-6\cos{4}A+15\cos2 A-20$
So we have,
$\sin^6 A=\frac{-1}{2^6}(2\cos 6 A-12\cos{4}A+30\cos2A-20)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
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Why does the tangent of numbers very close to $\frac{\pi}{2}$ resemble the number of degrees in a radian? Testing with my calculator in degree mode, I have found the following to be true:
$$\tan \left(90 - \frac{1}{10^n}\right) \approx \frac{180}{\pi} \times 10^n, n \in \mathbb{N}$$
Why is this? What is the proof or ex... | First, note that by standard trigonometric identities,
$$\tan\left(90^\circ-\frac{1^\circ}{10^n}\right)= \frac{\sin\left(90^\circ-\frac{1^\circ}{10^n}\right)}{\cos\left(90^\circ-\frac{1^\circ}{10^n}\right)}=\frac{\cos\left(\frac{1^\circ}{10^n}\right)}{\sin\left(\frac{1^\circ}{10^n}\right)}=\frac{\cos\left(\frac{1}{10^n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
$\epsilon$-$\delta$ proof that $\lim\limits_{x \to 1} \frac{1}{x} = 1$. I'm starting Spivak's Calculus and finally decided to learn how to write epsilon-delta proofs.
I have been working on chapter 5, number 3(ii). The problem, in essence, asks to prove that
$$\lim\limits_{x \to 1} \frac{1}{x} = 1.$$
Here's how I st... | $\tag 1 |\frac{1}{x} - 1| < \varepsilon \text{ and } 0 \lt \varepsilon \le \frac{1}{2}$
$ \text{ iff } \; -\varepsilon < \frac{1}{x} - 1 < \varepsilon \text{ and } 0 \lt \varepsilon \le \frac{1}{2}$
$ \text{ iff } \; 1 -\varepsilon < \frac{1}{x} < 1 + \varepsilon \text{ and } 0 \lt \varepsilon \le \frac{1}{2}$
$ \text{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "57",
"answer_count": 5,
"answer_id": 2
} |
Confirm the meaning of Prime and Primitive in a Galois(2) polynomial. Here it discusses primality (or more accurately irreducibility) and primitivity of polynomials in $G(2)$. More specifically it states that $x^6 + x + 1$ is irreducible and primitive.
But here I can divide $x^7 + 1$ by $x^6 + x + 1$ and get $x$ remain... | According to the definition in the first link you provided, to see that the order of
$x^6+x+1$ is $2^6-1$, you only need to check that $(x^6+x+1)\nmid (x^{n}+1)$ for all $n<2^6-1$, and that $x^6+x+1$ does divide $x^{2^6-1}+1$. A computer check easily confirm that this is the case.
I don't see any reason why the divisi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Indefinite integration : $\int \frac{1+x-x^2}{\sqrt{(1-x^2)^3}}$ Problem :
Solve : $\int \frac{1+x-x^2}{\sqrt{(1-x^2)^3}}$
I tried :
$\frac{1-x^2}{\sqrt{(1-x^2)^3}} + \frac{x}{\sqrt{(1-x^2)^3}}$
But it's not working....Please guide how to proceed
| What you did is a very good first step.
For the second integral, make the substitution $u=1-x^2$. We end up with the easy $\int -\frac{1}{2}u^{-3/2}\,du $.
For the first integral, note that the bottom simplifies to $(1-x^2)\sqrt{1-x^2}$, since the square root is only defined when $|x|\le 1$. There is nice cancellati... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
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Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$ I am trying to find a closed form for
$$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$
It seems that the answer is
$$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3}... | This isn't an answer to the question, but I thought I should post some of my work here.
Consider the function
$$
F(a) \;=\; \int_0^1 \frac{\log(1+x^a)}{1+x}dx.
$$
The question asks us to prove that $F(2+\sqrt{3}) = \dfrac{\pi^2}{12}(1+\sqrt{3}) + \log(2)\log(1+\sqrt{3})$.
*
*Mathematica isn't able to compute a clo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "169",
"answer_count": 3,
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Show that $e^x \geq (3/2) x^2$ for all non-negative $x$ I am attempting to solve a two-part problem, posed in Buck's Advanced Calculus on page 153. It asks "Show that $e^x \geq \frac{3}{2}x^2$ $\forall x\geq 0$. Can $3/2$ be replaced by a larger constant?"
This is after the section regarding Taylor polynomials, so I h... | Consider a quadratic $y=a x^2$ that is tangent to $y=e^x$. That is, we match function and derivative at a point $x=x_0$. Then
$$e^{x_0}=a x_0^2$$
$$e^{x_0}=2 a x_0$$
Then $a x_0^2=2 a x_0$ so that $x_0=2$. Plugging this back into one of the above equations, we get that
$$a=\frac{e^2}{4} \approx 1.84726$$
Consider, as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
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Finding Big-O with Fractions I'd want to know how I can find the lowest integer n such that f(x) is big-O($x^n$) for
a) $f(x) = \frac {x^4 + x^2 + 1}{x^3 + 1}$
I've fooled around with this a bit and tried going from
$\frac {x^4 + x^2 + 1}{x^3 + 1} \le \frac {x^4 + x^2 + x}{x^3 + 1} \le \frac {x^4 + x^2 + x}{x^3}$
but... | Intuitively, the idea is that the "order" of a function is that of its highest-order term. In $x^4 + x^2 + 1$, you can usually ignore everything except the $x^4$, and say that it is $O(x^4)$ and not $O(x^d)$ for any smaller $d < 4$. Similarly $x^3 + 1$ is of the order of $x^3$, and in general a polynomial of degree $d$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Showing that $\lim\limits_{n\to\infty}x_n$ exists, where $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}}$ Let $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}}$
a) Show that $x_{n} < x_{n+1}$
b) Show that $x_{n+1}^{2} \leq 1+ \sqrt{2} x_{n}$
Hint : Square $x_{n+1}$ and factor a 2 out of the square root
c) ... | 10 days old question, but .
a) Is already clear, that $ \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}} < \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n+1}}}}$ , because $\sqrt{n} <\sqrt{n} + \sqrt{n+1}$ which is trivial.
My point here is to give some opinion about b) and c), for me it's better to do the c) first. We know tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Repeated Eigenvalues Two problems from Differential Equations; Dynamical Systems, and an Introduction to Chaos (Morris W. Hirsch,Stephen Smale.Robert L. Devaney), examples page 112,113:
If $$A= \begin{pmatrix}
2 & 0 & -1 \\
0 & 2 & 1 \\
-1 & -1 & 2 \\
\end{pmatrix}$$
… $\lambda =2 , m_... | I will assume you understand how they derived $P$. (If not, please respond).
We have:
$$P= \begin{pmatrix}1& 0 & 1 \\-1 & 0 & 0 \\0 & -1 & 0 \end{pmatrix}$$
This give us:
$$P^{-1}= \begin{pmatrix} 0 & -1 & 0 \\0 & 0 & -1 \\1 & 1 & 0 \end{pmatrix}$$
From this we get:
$$P^{-1}AP = \begin{pmatrix} 0 & -1 & 0 \\0 & 0 & -1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$ Problem statement:
Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $.
, $n\in \mathbb{N}$
My progress
LHS is e... | Suppose $$\frac{2(k+1)-1}{2(k+1)}.\frac{1}{\sqrt{3k+1}}\geq \frac{1}{\sqrt{3(k+1)+1}}$$
$$ \frac{2k+1}{2(k+1)}.\frac{1}{\sqrt{3k+1}}\geq \frac{1}{\sqrt{3(k+1)+1}}$$
$$\frac{(2k+1)^2}{4(k+1)^2 (3k+1)}\geq \frac{1}{3(k+1)+1}$$
$$ 3(k+1)(2k+1)^2+(2k+1)^2 \geq 4(k^2+2k+1)(3k+1)$$
$$ (3k+3)(4k^2+4k+1)+4k^2+4k+1 \geq (k^2+2k... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 6
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If $A = \tan6^{\circ} \tan42^{\circ},~~B = \cot 66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$ My trigonometric problem is:
If $A = \tan6^{\circ} \tan42^{\circ}$ B = cot$66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$.
Working :
$$B = \cot 66^{\circ} \cot78^{\circ} = 1- \frac{\tan... | Using
$2\cos A\cos B=\cos(A-B)+\cos(A+B)$ and $2\sin A\sin B=\cos(A-B)-\cos(A+B),$
$$A=\frac{\sin 6^\circ\cdot \sin 42^\circ}{\cos 6^\circ\cdot \cos 42^\circ}=\frac{\cos36^\circ-\cos48^\circ}{\cos36^\circ+\cos48^\circ}$$
Applying Componendo and dividendo,
$$\frac{1+A}{1-A}=\frac{\cos36^\circ}{\cos48^\circ}$$
Similarly... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
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A function inequality Is it true that the following function
$$\frac{\pi ^2 \left(t^2-4 (-1+t) \text{cos}\left[\frac{\pi }{m}\right]^2\right) \text{csc}\left[\frac{\pi (-2+t)}{m}\right]^2}{m^2}, t\in[0,1]$$ attains its maximum in 0 and 1. Here $m>3$.
| Let us denote $$f(t) := \left(t^2-4(-1+t)\cos^2\left(\frac \pi m \right)\right)\csc^2 \left(\frac {\pi(-2+t)} m \right) .$$ We start from the verification $$f(0)=f(1)= \csc^2\left( \frac \pi m \right) . $$ Next, in order to prove the statement under consideration, it is enough to prove the convexity of $f(t)$ on $(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding limit function $\lim_{n \rightarrow \infty} n ((x^2 +x + 1)^{1/n} -1)$ \begin{align}
f(x) &= \lim_{n \rightarrow \infty} n ((x^2 +x + 1)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n ((\infty)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n (1 -1)\\& =
\lim_{n \rightarrow \infty} n \cdot 0 \\&= 0
\end{align}
Did... | Note that $$\lim\limits_{n \rightarrow \infty} n ((x^2 +x + 1)^{\tfrac{1}{n}} -1) =\lim\limits_{n\rightarrow \infty} \dfrac{(x^2 +x + 1)^{\tfrac{1}{n}} -1}{\dfrac{1}{n}}. $$
Denoting $h(x)=x^2 +x + 1,$ we have
$$\lim\limits_{n\rightarrow \infty} \dfrac{(x^2 +x + 1)^{\tfrac{1}{n}} -1}{\dfrac{1}{n}}=\lim\limits_{n\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Solve $\frac{dx}{dt} = x^3 + x$ for $x$ This is a seemingly simple first order separable differential equation that I'm getting stuck on. This is what I have so far:
$$\frac{dx}{dt} = x^3+x$$
goes to
$$\frac{dx}{x(1+x^2)} = dt$$
Now using partial fractions to integrate the left-hand side:
$$\frac{1}{x(1+x^2)} = \frac{... | Assume that $x\ge 0$.
$$\frac{x}{(1+x^2)^{\frac{1}{2}}}= \frac{\sqrt{x^2}}{(1+x^2)^{\frac{1}{2}}}=\sqrt{\frac{x^2}{1+x^2}}$$
and you can probably take it from there.
If $x<0$, then $x=-\sqrt{x^2}$ and proceed similarly.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/433966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
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Plotting graphs using numerical/mathematica method From the author's equation 13, 14 We can write by inserting V''(A)=0,
Solving for R we get,
$$R= \frac{6^{D/4} \sqrt{D}}{\sqrt{-2^{1+\frac{D}{2}} 3^{D/2}+3 2^{1+D} A-3^{1+\frac{D}{2}} A^2}}$$
Now inserting the V into the article equation (11)$$E= \left(\frac{\pi }{2... | Look at your function in a much more simple way:
$$R(A) = \frac{\sqrt{12}}{\sqrt{24 A - 9 A^2-12}}$$
for $D=2$. The function should have a horizontal tangent when the derivative of the radicand is zero:
$$\frac{d}{dA} (24 A - 9 A^2-12) = 0 \implies A = \frac{4}{3}$$
which seems to agree with your result. Also, $R$ bl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/434768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Partial fractions to integrate$\int \frac{4x^2 -20}{(2x+5)^3}dx$ $$\int \frac{4x^2 -20}{(2x+5)^3}dx$$
I can't use the coverup method that I learned since making anything zero in this makes everything zero. I would probably just use random test points because I don't have any other tricks memorized. Is there some specif... | $$\int \frac{4x^2 -20}{(2x+5)^3}dx = \int \dfrac A{(2x + 5)} + \dfrac B{(2x + 5)^2} + \dfrac C{(2x + 5)^3}\, dx$$
Partial fractions can be thought of as the reverse of "finding the common denominator".
That is, we can equate the numerator of the original integrand with the expansion of each of the numerators of the des... | {
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"url": "https://math.stackexchange.com/questions/437153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all real numbers such that $\sqrt{x-\frac{1}{x}} + \sqrt{1 - \frac{1}{x}} = x$ Find all real numbers such that
$$\sqrt{x-\frac{1}{x}} + \sqrt{1 - \frac{1}{x}} = x$$
My attempt to the solution :
I tried to square both sides and tried to remove the root but the equation became of 6th degree.Is there an easier metho... | Clearly, $x\ne0$
$$\sqrt{x-1/x} + \sqrt{1 - 1/x} = x$$
$$\implies \sqrt{x^2-1}+\sqrt{x-1}=x\sqrt x$$
$$\implies \sqrt{x-1}(\sqrt{x+1}+1)=x\sqrt x$$
$$\implies \sqrt{x-1}\frac{x+1-x}{\sqrt{x+1}-1}=x\sqrt x\text{ (rationalizing the numerator) }$$
$$\implies \sqrt{x-1}=\sqrt x(\sqrt{x+1}-1)=\sqrt{x^2+x}-\sqrt x$$
$$\impli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
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Using complete induction, prove that if $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$, then $a_n=2^n$ Could anyone please explain to me how to do this problem by using the principle of complete induction? Thanks. :)
Let $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$ for all $n\geq 1$. Prove that $a_n=2^n$ for all natur... | For induction we wish to show the following:
*
*If $a_{n+2} = 2^{n+2}$ and $a_{n+1} = 2^{n+1}$ for some particular positive integer $n$ then: $a_{n+3} = 2^{n+3}$ must be true for that same value $n$
*$a_1 = 2^1$ and $a_2 = 2^2$
Since if 1 is true and 2 is true then we can repeatedly use 1 from the truth of 2 to est... | {
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Solving a linear system with complex eigenvalues I have the system:
\begin{equation}
x' = \begin{pmatrix}5&10\\-1&-1\end{pmatrix}x
\end{equation}
The corresponding characteristic equation is:
\begin{equation}
\lambda^2-4\lambda+5 \\
\implies \lambda_1 = 2+i \land \lambda_2 = 2-i
\end{equation}
I am having trouble solvi... | If we set $x_1 = (a, b)^T$, we get
$$
\begin{pmatrix}3-i&10\\-1&-3-i\end{pmatrix}x_1 = \begin{pmatrix}3-i&10\\-1&-3-i\end{pmatrix}\begin{pmatrix}a\\b\end{pmatrix}\\\\
= \begin{pmatrix}(3-i)a + 10b\\-a - (3 + i)b\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}
$$
which is just a set of two equations of two (complex) unk... | {
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what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by 47? Can any one please tell the approach or solve the question
what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by $47$?
I can solve remainder of $45!$ divided by $47$ using Wilson's theorem but I don't know what must be the approach for... | Just to compose table:
\begin{array}{|c|r|}
\hline
n! & \equiv \ldots (\bmod \:47) \\
\hline \\
1! & 1 \\
2! & 2\cdot 1 = 2 \\
3! & 3 \cdot 2 = 6 \\
4! & 4 \cdot 6 = 24 \\
5! & 5 \cdot 24 = 120 \equiv 26 \\
6! & 6 \cdot 26 = 156 \equiv 15 \\
7! & 7 \cdot 15 = 105 \equiv 11 \\
\cdots \\
44! & 44 \cdot 8 = 352 \equiv 23... | {
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Saddle Point Integral I want to calculate ,
$$I = \int_0^\infty dx \,x^{2n}e^{-ax^2 -\frac{b}{2}x^4} $$
for real positive a, b and positive integer n. n is the large parameter. Using Saddle Point Integration
I find saddle points by setting the derivative P'(x) = 0 where
$$ P(x) = n\log(x^2) -ax^2 -\frac{b}{2}x^4$$
I... | $\int_0^\infty x^{2n}e^{-ax^2-\frac{b}{2}x^4}~dx$
$=\int_0^\infty x^{2n}e^{-x^2\left(a+\frac{b}{2}x^2\right)}~dx$
$=\int_0^\infty\left(\dfrac{\sqrt{2a}\sinh x}{\sqrt{b}}\right)^{2n}e^{-\left(\frac{\sqrt{2a}\sinh x}{\sqrt{b}}\right)^2\left(a+\frac{b}{2}\left(\frac{\sqrt{2a}\sinh x}{\sqrt{b}}\right)^2\right)}~d\left(\dfr... | {
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If $x,y,z>0$ are distinct and $x+y+z=1$ what is the minimum of $\left((1+x)(1+y)(1+z)\right)/\left((1-x)(1-y)(1-z)\right)$? If $x,y,z>0$ are not equal and positive and if $x+y+z=1$ the expression
$$\frac{(1+x)(1+y)(1+z)}{(1-x)(1-y)(1-z)}$$
is greater than what quantity?
| By AM-GM we obtain:
$$\frac{\prod\limits_{cyc}(1+x)}{\prod\limits_{cyc}(1-x)}=\frac{\prod\limits_{cyc}(2x+y+z)}{\prod\limits_{cyc}(x+y)}=\frac{\prod\limits_{cyc}(x+y+x+z)}{\prod\limits_{cyc}(x+y)}\geq\frac{8\prod\limits_{cyc}\sqrt{(x+y)(x+z)}}{\prod\limits_{cyc}(x+y)}=8.$$
The equality occurs for $x=y=z=\frac{1}{3}$, w... | {
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If $a,b,c > 0$ satisfy $a^2+b^2+c^2=3$ then $\frac{a}{b+c+3}+\frac{b}{a+c+3}+\frac{c}{a+b+3} \geq \frac{3}{5}$
Given $a,b,c > 0$ satisfying the condition $$a^2+b^2+c^2=3,$$
prove that
$$\frac{a}{b+c+3}+\frac{b}{a+c+3}+\frac{c}{a+b+3} \geq \frac{3}{5}.$$
Thank you all
| This can be done automagically using Lagrange multipliers (like most inequalities of this kind...)
Take $f=\frac{a}{b+c+3}+\frac{c}{a+b+3}+\frac{b}{a+c+3}$, $g=a^2+b^2+c^2$, and find the critical points of $f$ subject to the condition that $g=3$. In Mathematica, this can be done as follows:
In[15]:= f = a/(b + c + 3) +... | {
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} |
An identity which applies to all of the natural numbers Prove that any natural number n can be written as $$n=a^2+b^2-c^2$$ where $a,b,c$ are also natural.
| Consider $n\ge 6$.
If $n$ is odd, $n=2m+1$, then
$$
n = 2m+1 = 2^2 + (m-1)^2 - (m-2)^2;
$$
If $n$ is even, $n=2m$, then
$$
n = 2m = 1^2 + m^2 - (m-1)^2;
$$
Small $n$:
$1=1^2+1^2-1^2$,
$2=3^2+3^2-4^2$,
$3=4^2+6^2-7^2$,
$4=2^2+1^2-1^2$,
$5=4^2+5^2-6^2$.
| {
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Show that $\frac {\sin(3x)}{ \sin x} + \frac {\cos(3x)}{ \cos x} = 4\cos(2x)$ Show that
$$\frac{\sin(3x)}{\sin x} + \frac{\cos(3x)}{\cos x} = 4\cos(2x).$$
| We have,
$$\sin(3x) = 3\sin(x) - 4\sin^{3}(x)$$
and
$$\cos(3x) = 4\cos^{3}(x) - 3 \cos(x).$$
Thus, your expression reduces to
$$3 - 4\sin^{2}(x) + 4\cos^{2}(x) - 3$$
which is equal to
$$4 \cos(2x),$$
as $\cos(2x) = \cos^{2}(x) - \sin^{2}(x)$.
| {
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"answer_id": 3
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How to find the limit $x_n=\frac{(a)(a+1)(a+2)..(a+n)}{(b)(b+1)(b+2)..(b+n)}$ Let a,b be positive number then how to find the limit of $x_n=\frac{(a)(a+1)(a+2)..(a+n)}{(b)(b+1)(b+2)..(b+n)}$ when $n\rightarrow \infty $ when a=b obviously the limit is 1 but what about $a<b$ ?
this problem reminded me of the limit $0<A_n... | (Since $x_n>x_{n+1}>0$ $\lim_{n\to\infty} x_n$ does exist and $\geq0$.) Applying the inequality $1+x\leq e^x$ (which is valid for any real $x$) we obtain
$$
\frac{a+k}{b+k}=\frac{b+k+a-b}{b+k}=1+\frac{a-b}{b+k}\leq e^{\frac{a-b}{b+k}}.
$$
Using this inequality we obtain
$$
0\leq x_n=\frac{(a)(a+1)(a+2)..(a+n)}{(b)(b+1)... | {
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Prove that $(a-d)^2+(b-c)^2\geq 1.6$ if $a^2+4b^2=4$ and $cd=4$ Let $a, b, c, d$ be real numbers. I need to prove this innocent inequality
$$
(a-d)^2+(b-c)^2\geq 1.6
$$
if $a^2+4b^2=4$ and $cd=4$.
I was told that there exist nice and sweet elementary solutions.
| Using $a^2 + 4b^2 = 4$ and $cd = 4$, we have
\begin{align*}
&(a - d)^2 + (b - c)^2 + \frac12(a^2 + 4b^2 - 4) - \frac{2\sqrt 2}{3}(cd - 4)\\[6pt]
=\,& \frac16(3a - 2d)^2 + \frac13(3b - c)^2 + \frac13(d - c\sqrt 2)^2 + \frac{8\sqrt 2}{3} - 2 \\[6pt]
\ge\, &\frac{8\sqrt 2}{3} - 2\\
>\,& \frac85.
\end{align*}
We are do... | {
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"answer_count": 4,
"answer_id": 3
} |
Maclaurin series of $f(x) = e^x \sin x$ $$f(x) = e^x \sin x$$
I tried applying the given formula in my book but it didn't work.
The maclaurin for $e^x$ is given as $\displaystyle \sum \frac{x^n}{n!}$ and $\sin x$ $\displaystyle \sum \frac{(-1)^n x^{2n + 1}}{(2n+1)!}$
I attempted to multiply them together, failed teh bo... | It’s just like multiplying polynomials.
Just as
$$(a+b+c)(d+e+f)=ad+ae+af+bd+be+bf+cd+ce+cf$$
is the sum of all possible products of one term from the first factor and one term from the second factor, so also is the product
$$\left(1+x+\frac{x^2}2+\frac{x^3}6+\ldots\right)\left(x-\frac{x^3}6+\frac{x^5}{120}-\frac{x^7... | {
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"answer_count": 7,
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Triple integral (check solution) The function given is
$f(x,y,z) = \displaystyle\frac{1}{(x+y+z+1)^2}$
$D = \{(x,y,z) \in \mathbb{R}^3 : x \geq 0, y \geq 0, z \geq 0, x+y+z \leq 1 \}$
It seems that the domain would be the under a plane that contains the points $(1,0,0)$, $(0,1,0)$, $(0,0,1)$; then if i consider the pla... | I don't think so. I think the integral looks like
$$\int_0^1 dx \, \int_0^{1-x} dy \, \int_0^{1-x-y} dz \frac{1}{(1+x+y+z)^2}$$
which is equal to
$$-\int_0^1 dx \, \int_0^{1-x} dy \, \left [\frac{1}{2} - \frac{1}{1+x+y} \right ]$$
which is equal to
$$-\int_0^1 dx \, \left [\frac12 (1-x) - \log{2} + \log{(1+x)} \right ... | {
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Show that $\sin2\alpha\cos\alpha+\cos2\alpha\sin\alpha = \sin4\alpha\cos\alpha - \cos4\alpha\sin\alpha$
Show that $\sin2\alpha\cos\alpha+\cos2\alpha\sin\alpha = \sin4\alpha\cos\alpha - \cos4\alpha\sin\alpha$
I know that $\sin2\alpha = 2\sin\alpha\cos\alpha$
so
$$\sin2\alpha\cos\alpha=2\sin\alpha\cos^2\alpha$$
and $\... | use $$\sin(A+B) = \sin A\cos B + \cos A\sin B $$on LHS and
$$\sin(A-B) =\sin A\cos B - \cos A\sin B$$ on RHS
so $$\sin(3\alpha) = \sin(3\alpha)$$
| {
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Conversion of sum of series into product form: $\sum_{n=1}^\infty \frac1{n(n+1)} = \frac12 \prod_{n=2}^\infty \left( 1+\frac{1}{n^2-1} \right)$ Show that the following series and product are equivalent:
$$
\sum_{n=1}^\infty \left[ \dfrac{1}{n(n+1)} \right] = \dfrac{1}{2} \prod_{n=2}^\infty \left[ 1+\dfrac{1}{n^2-1} \r... | $\frac 1{n(n+1)}= \frac1n-\frac1{n+1}$ so this is a telescoping sum, almost all terms will cancel out:
$$ \sum_{n=1}^N \frac1{n(n+1)}= 1 - \frac12 + \frac12 - \frac13 + \dots + \frac1{N-1}-\frac1N + \frac1N-\frac1{N+1}=1-\frac1{N+1}.$$
Similarly, you can see that in the product of the terms of the form $\frac{n^2}{n^2-... | {
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Gauge fractions with exponents - No Calculator How does one (without the use of Calculator) determine that $5/6$ is less than $(35/36)^6$? How is this done mentally?
| If you look at your second number, it can be written as
$$( 1 - \frac{1}{36} )^6 \ = \ 1 \ - \ 6 \cdot 1^5 \cdot \frac{1}{36} \ + \ \binom 62 \cdot 1^4 \cdot (\frac{1}{36})^2 \ - \ ... $$
with the remaining (unwritten) terms being very small (the first two terms equal $ \ \frac{5}{6} \ $ ) . So $ \ \frac{5}{6} \ ... | {
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Difficult Gaussian Integral Involving Two Trig Functions in the Exponent: Any Help? Here's the integral:
$$\int_d^e \exp\left(-a\left((b+c)\cos(x)-\sqrt{b^2 - (b+c)^2 \sin^2(x)}\right)^2 \right) \, dx$$
I've tried using Mathematica: it fails.
Can anyone help evaluate it?
Perhaps some kind of series expansion of the e... | Assume $b\neq0$ , $c\neq0$ and $b+c\neq0$ to maintain the key meaning of the question.
$\int_d^e\exp\left(-a\left((b+c)\cos x-\sqrt{b^2-(b+c)^2\sin^2x}\right)^2 \right)~dx$
$=\int_d^e\sum\limits_{n=0}^\infty\dfrac{(-1)^na^n\left((b+c)\cos x-\sqrt{b^2-(b+c)^2\sin^2x}\right)^{2n}}{n!}dx$
$=\int_d^e\sum\limits_{n=0}^\inft... | {
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A further question on the irrationality of $x^2+y^2=3$ (Apologies for a further question on the same problem)
On page 79 of Julian Harvil's book "The Irrationals" he sets out to prove (by contradiction) that all the points on the circle described by $x^2+y^2=3$ are irrational.
To paraphrase his proof:
*
*Let $\left(\... | Remember that the square $c^2$ of an integer $c$ is either $\equiv 0\pmod 4$ (if $c$ is even) or $\equiv 1\pmod 8$ (if $c$ is odd); in fact $1\pmod 4$ instead of $1\pmod 8$ is good enough here, as we obtain $3c^2\equiv 0\pmod 4$ or $3c^3\equiv 3\pmod 4$, but definitelys not $3c^3\equiv 1\pmod 4$.
| {
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What is the probability of getting a sum of 7 or at least one 5 when you roll two die Please tell me how to approach this problem.
(Sum of 7) = {4+3, 3+4, 6+1, 1+6, 5+2, 2+5} = 6
(At Least one 5) = {1+5, 2+5, 3+5, 4+5, 6+5, 5+1, 5+3, 5+4, 5+5, 5+6} = 10
so the answer will be 16/36 = 4/9 ?
| You double counted some possibilities (the two events are not mutually exclusive!). By taking the union of the sets of desirable outcomes, we obtain:
$$ \left\{\begin{align*}
&(1,5),(2,5),(3,5),(4,5),(5,5),(6,5),\\
&(5,1),(5,2),(5,3),(5,4),~~~~~~~~~~~(5,6),\\
&(1,6),(6,1),(3,4),(4,3)~~~~~~~~~~~~~~~~~~~~~~~
\end{align*}... | {
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"answer_count": 2,
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Initial value problem differential equation $y' = (x-1)(y-2)$ $$y' = (x-1)(y-2)$$
$y(2)= 4$
$$\frac{1}{y-2}dy = (x-1)dx$$
$$\int \frac{1}{y-2}dy =\int (x-1)dx$$
$$\ln(y-2) = \frac{x^2}{2} - x + c$$
$$y - 2 = e^{\frac{x^2}{2} - x + c} $$
$$y = e^{\frac{x^2}{2} - x + c} + 2$$
plug in the inital value
$$y = e^{\frac{2^2}{... | Continuing from your last step:
$$
y = e^c + 2 = 4\\
\Rightarrow c = \ln2
$$
finally giving
$$
y = e^{\frac{1}{2}x^2-x+\ln2} +2
$$
or equivalently,
$$
y = 2e^{\frac{1}{2}x^2-x} +2
$$
| {
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"answer_count": 3,
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Convergence of sum related to $\sum_k 1/p_k$ Motivation (skip if desired): We can show that $$S =\frac{1}{2}+ \frac{1}{2\cdot 3}+ \frac{(2)}{2\cdot (3)\cdot5}+\frac{(2\cdot 4)}{2\cdot(3\cdot5)\cdot 7}+\frac{(2\cdot4\cdot6)}{2\cdot(3\cdot5\cdot7)\cdot11}+...$$etc. converges.
We know
$$\frac{1}{2}\sum_{k=1}\frac{1}{p_k}... | As the commenters have suggested: Mertens proved the asymptotic formula
$$
\prod_{p\le x} \bigg( 1-\frac1p \bigg) \sim \frac{e^{-\gamma}}{\log x}
$$
(where $\gamma$ is Euler's constant). Therefore
$$
\prod_{k=3}^{n} \frac{p_{k-1}-1}{p_{k-1}} = 2 \prod_{p\le p_{n-1}} \bigg( 1-\frac1p \bigg) \sim \frac{2e^{-\gamma}}{\log... | {
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"answer_count": 1,
"answer_id": 0
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Proving that a discrete stochastic variable is binomial distributed. Given a discrete stochastic variables, with the probability function;
$$
p_{X}\left(x\right)=\left\{
\begin{array}{cc}
\frac{1}{4} & \text{if } x = -1 \\
\frac{1}{4} & \text{if } x = 0 \\
\frac{1}{2} & \text{if } x = 1 \\
0 & \text{otherwise}
\end{arr... | You already did all of the work. You showed that
$$
p_{\left|\mathbf{X}\right|}\left(x\right)
=
\left(\begin{array}{c}
1\\
x
\end{array}\right)\frac{3}{4}^{x}\frac{1}{4}^{1-x}
=
\begin{cases}
\frac{3}{4} & \text{if }x=1\\
\frac{1}{4} & \text{if }x=0
\end{cases}
$$
| {
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"answer_id": 0
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Find $a$ such that $P(X\le a)=\frac 1 2$
Given probability density function $f(x)=\begin{cases} 1.5(1-x^2),&0<x<1\\0& \mbox{otherwise}\end{cases}$, calculate for which '$a$', $P(X\le a)=\frac 1 2$ (the solution is $2\cos\left(\frac{4\pi}{9}\right))$.
I calculated the integral and got $P(X\le a)=\displaystyle\int_0^af... | All you should do is just finding roots of the following polynomial which lie within $[0,1]$
$$
1.5a\left(1-\frac {a^2}{3}\right)-\frac 1 2 \\
\Rightarrow a^3-3a+1
$$
Now you can solve it using methods available for solving cubic polynomials or you can write it in this way :
$$
\begin{align}
a^3-3a+1 &= (a-x_1)(a-x_2)(... | {
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Some identities with the Riemann zeta function Can someone either help derive or give a reference to the identities in Appendix B, page 27 of this, http://arxiv.org/pdf/1111.6290v2.pdf
Here is a reproduction of Appendix B from Klebanov, Pufu, Sachdev and Safdi's $2012$ preprint (v2) 'Renyi Entropies for Free Field Theo... | $(B.1)$ is the functional equation of the Hurwitz zeta function and Knopp and Robins' proof is available here.
$$\tag{B.1}\zeta(z,a)=\frac{2\,\Gamma(1-z)}{(2\pi)^{1-z}}\left[\sin\frac {z\pi}2\sum_{n=1}^\infty\frac{\cos2\pi an}{n^{1-z}}+\cos\frac {z\pi}2\sum_{n=1}^\infty\frac{\sin2\pi an}{n^{1-z}}\right]$$
The author of... | {
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Prove the Inequality: $\sum\frac{x^3}{2x^2+y^2}\ge\frac{x+y+z}{3}$ Let $x, y, z>0$. Prove that:
$$\frac{x^3}{2x^2+y^2}+\frac{y^3}{2y^2+z^2}+\frac{z^3}{2z^2+x^2}\ge\frac{x+y+z}{3}$$
| I have a direction which may be fruitful but didn't work out all cases. Rewrite the inequality as:
$$
\frac{x^3}{2x^2+y^2}-\frac{x}{3}+\frac{y^3}{2y^2+z^2}-\frac{y}{3}+\frac{z^3}{2z^2+x^2}-\frac{z}{3}\ge0
$$
Let $f(t)=\frac{1-t^2}{2+t^2}$, then the inequality is equivalent to: $xf(a)+yf(b)+zf(c)\ge0$, where $a=y/x, b=z... | {
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} |
A first order non-linear ordinary differential equation How do we solve the first order non-linear ordinary differential equation ,
$$y^3=xy^2 \frac{dy}{dx} + x^4 \Big(\frac{dy}{dx}\Big)^2$$
| Firstly we note that: $$y(x)=0 \tag{Solution 1}$$ is a solution, hereafter we take $y(x)\ne0$.
Substitute: $$y(x)=\dfrac{1}{h(x)}$$ into:$$ y \left( x \right) ^{3}=x y \left( x \right)
^{2}{\frac {d}{dx}}y \left( x \right) +{x}^{4} \left( {\frac
{d}{dx}}y \left( x \right) \right) ^{2}\tag{1}$$
to get:
$$\begin{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Inequality related to the continued fraction expansion of $\sqrt 3$ I am working on a problem related to the continued fraction expansion of $\sqrt 3$. If $p_k$ and $q_k$ denote the numerator and denominator, respectively, of the $k$th convergent, I should show that
$$\left|\sqrt{3}-\frac{p_{2n+1}}{q_{2n+1}}\right| \lt... | From the period length of $2$, you obtain that for all $n$, you have
$$p_{2n+1}^2 - 3q_{2n+1}^2 = 1,$$
and from that you can deduce
$$\begin{align}
\frac{p_{2n+1}}{q_{2n+1}} - \sqrt{3} &= \frac{p_{2n+1} - \sqrt{3}q_{2n+1}}{q_{2n+1}}\\
&= \frac{(p_{2n+1} - \sqrt{3}q_{2n+1})(p_{2n+1} + \sqrt{3}q_{2n+1})}{q_{2n+1}(p_{2n+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A $4$ variable inequality If $a,b,c,d$ are positive numbers such that $c^2+d^2=(a^2+b^2)^3$, prove that
$$\frac{a^3}{c} + \frac{b^3}{d} \ge 1,$$
with equality if and only if $ad=bc$.
Source: Don Sokolowsky, Crux Mathematicorum, Vol. 6, No. 8, October 1980, p.259.
| Because by Holder
$$\left(\frac{a^3}{c}+\frac{b^3}{d}\right)^2(c^2+d^2)\geq(a^2+b^2)^3$$
and we are done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/471739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Probability of first and second drawn balls of the same color, without replacement I have an urn with 10 balls: 4 red and 6 white. What is the probability that the first two drawn balls have the same color?
Approach No. 1:
$|U| = 10! = 3628800$
White balls on first and second draw: on the first and second draw, I can p... | S = {Set of all possible combination of 2 balls from the available 4 Red and 6 White Balls}
n(S) = $^{10} P_{2}$ ways = $10 \times 9 = 90$ (Since, 2 balls can be selected from $10$ in $^{10} P_{2}$ ways)
Now,
E = Both are of same colour
n(E) = $2$ Red balls can be selected from $4$ in $^4 P_{2}$ ways AND $2$ White ball... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/472819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Determine the minimum of $a^2 + b^2$ if $a,b\in\mathbb{R}$ are such that $x^4 + ax^3 + bx^2 + ax + 1 = 0$ has at least one real solution I just wanted the solution, a hint or a start to the following question.
Determine the minimum of $a^2 + b^2$ if $a$ and $b$ are real numbers for which
the equation
$$x^4 + ax^3 + bx^... | [The following solution relies on the ideas in the previous two answers.]
Let the roots of $x^4+ax^3+bx^2+ax+1=0$ be given by $c, \frac{1}{c}, d, \frac{1}{d}$ where $c\in\mathbb{R}$, and
let $t=c+\frac{1}{c}$ and $l=d+\frac{1}{d}$.
Then $|t|\ge2$ since $c+\frac{1}{c}\ge2$ if $c>0$ and $c+\frac{1}{c}\le2$ if $c<0$.
S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/474507",
"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.