A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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1answer
27 views

Fourier coefficient and computing an improper integral

I am having difficulties with this problem. I don't really know where to start, I suspect there is something I am supposed to know or "see" that I am missing. $u$ is a $2\pi$-periodic function ...
2
votes
1answer
49 views

Generating function for $\sum_{n=0}^{\infty} n^k x^n$

I would like to get the closed form for this generating function, assuming the $k$ is given up front / held constant: $\sum_{n=0}^{\infty} n^k x^n$ However I don't know if this is too advanced for me ...
2
votes
2answers
90 views

How to evaluate $\int\frac{dx}{(2\sin x+\sec x)^4}$?

I tried a lot but I am not able to get a start. Can anyone give me the start of this question $$ \int\frac{dx}{(2\sin x+\sec x)^4} \ ? $$
2
votes
3answers
79 views

Solving an equation involving an integral: $\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$

Determine a pair of number $a$ and $b$ for which $$\int_0^1\frac{ax+b}{(x^2+3x+2)^2}\:dx=\frac52.$$ I tried putting $x$ as $1-x$ as the integral wouldn't change but could not move forward from ...
0
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1answer
53 views

Formula for number of faces in 4 dimensions

If a polytope has $m$ faces in 3 dimensions, how many faces does its analogous polytope have in four dimensions? Does a formula exist? For example, if $m=4$, you have a tetrahedron, and the 4-...
4
votes
4answers
101 views

Sum to infinity of trignometry inverse: $\sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right)$

If we have to find the value of the following (1) $$ \sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right) $$ I know that $$ \arctan \left(\frac{4}{r^2+3} \right)=\arctan \left(\frac{r+1}2 \right)-\...
7
votes
5answers
491 views

Trigonometry Olympiad problem: Evaluate $1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$

Find the value of $$1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$$ My attempt I converted the $\sin$ functions which have arguments greater than $90^\circ$ to $\...
0
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1answer
44 views

Path length of Gaussian

I am trying to find the path length of a Gaussian $f(x)=e^{-x^2/a^2}$ from $x=0$ to some positive point $x_0$. I've tried this by integrating the differential length, $ds^2=dx^2+dy^2$, but getting ...
7
votes
2answers
195 views

Value of this convergent series: $\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$

What is the value of- $$\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$$ I wrote it as general term $\sum\frac{n}{(2n+1)!}$. As the series converges it should be telescopic (my thought). But ...
1
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2answers
61 views

Proportionally Distributing $N$ items across $B$ bins.

My question is similar to this: Proportional Distribution My problem follows: I have $N$ items that cannot be broken up into fractional components, but should be distributed across $B$ bins where ...
0
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2answers
68 views

Maxwellian integral : is there a closed form?

$f_A(x,y)=\int_0^\infty du \frac{u \left(e^{-\frac{(u-x)^2}{2 A}}-e^{-\frac{(u+x)^2}{2 A}} \right)}{\sqrt{2 \pi } \sqrt{A} x \left(y^2+u^2\right)} $ is there a closed form? I was able to find ...
5
votes
4answers
122 views

How do we show that $\ln{2}-\gamma=\sum_{n=1}^{\infty}{\zeta(2n+1)\over 2^{2n}(2n+1)}?$

$$\ln{2}-\gamma=\sum_{n=1}^{\infty}{\zeta(2n+1)\over 2^{2n}(2n+1)}\tag1$$ Any hints?
3
votes
1answer
69 views

Another way to evaluate $\int\frac{\cos5x+\cos4x}{1-2\cos3x}{dx}$?

What I've done is this:$$\int\dfrac{\cos5x+\cos4x}{1-2\cos3x}{dx}$$ $$\int \dfrac{\sin 3x}{\sin 3x}\left[\dfrac{\cos5x+\cos4x}{1-2\cos3x}\right]{dx}$$ $$\dfrac {1}{2}\int\dfrac{\sin 8x -\sin 2x +\sin ...
7
votes
2answers
271 views

Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured

By trial and error I have found numerically $$\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}=\frac{1}{6}-\frac{1}{2\pi}$$ how can this result be derived analytically?
1
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2answers
76 views

Is there a closed form for this binomial sum?

I am looking for a closed form of this sum:$\sum\limits_{j=k}^n\binom{j}{k}(-1)^j$ I know that this sum has a closed form: $\sum\limits_{j=k}^n\binom{j}{k}=\binom{n+1}{k+1}$ I can get this closed ...
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votes
1answer
43 views

A question on the Laplace Transform of $f(t)=t e^{at}\sin (bt)$ [closed]

I would like to solve the Laplace transform of the following function: $$t \mapsto t e^{at}\sin (bt).$$ I know that $\mathscr{L}\left(\sin(bt)\right)=\dfrac{b}{s^2+b^2}$ and that you have to ...
8
votes
0answers
146 views

Closed form of $\int_0^1 \tan(\gamma\sqrt{1-x^2}) dx$

Some context: I'm studying the problem of nonperturbative pair creation from strong fields in quantum electrodynamics. For certain time dependent electric fields I can get some information about the ...
23
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3answers
472 views

The entry-level PhD integral: $\int_0^\infty\frac{\sin 3x\sin 4x\sin5x\cos6x}{x\sin^2 x\cosh x}\ dx$

I hope you find this integral interesting. Evaluate $$\int_0^\infty\frac{\sin\left(\,3x\,\right)\sin\left(\,4x\,\right) \sin\left(\,5x\,\right)\cos\left(\,6x\,\right)}{x\,\sin^{2}\left(\,x\,\...
1
vote
1answer
45 views

Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $

As the question says, $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$ where a is a constant, $a>0$.
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0answers
103 views

Summation of $\sum_{n=0}^\infty e^{-\sqrt n}$

Is there a closed form for the following sum? $$\sum_{n=0}^\infty e^{-\sqrt n}$$
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3answers
90 views

How to prove $\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right)$?

I need help, on how to prove $$\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right).$$ Any hints?
3
votes
1answer
44 views

$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, for complex variable $z$.

I want to find this limit for complex variable $z$ $$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$$ In the real case I know $\sin(z)$ is bounded by $-1, 1,$ and the limit is $0$. But in the complex case ...
1
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1answer
56 views

Evaluating $\int_{s^{-1/n}}^{\infty}v^2\exp{\left[-\left(\frac{l}{v} + m v\right)^2\right]}dv$

I am trying to evaluate the following $$I = \int_0^s u^{-3n-1} \exp{\left[-\left(l u^n + \frac{m}{u^{n}}\right)^2\right]}\,du,$$ where $l, m$ and $n$ are positive constants. I tried to substitute $v ...
1
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1answer
29 views

Closed Form Expressions: Summation and Product Operators [closed]

My question is: are expressions utilizing summation, $\Sigma$, and product, $\Pi$, operators considered 'closed-form'? To be more precise if the bounds in our summation/product operators contain ...
2
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4answers
80 views

Help with the integral $\int x\sqrt{\frac{1-x^2}{1+x^2}}dx$

I would like to know what is $$\int x\sqrt{\frac{1-x^2}{1+x^2}}dx.$$ I put $x=\tan(y)$ to get integral of $\displaystyle \int \frac{\sin(y)}{\cos^3(y)}.\sqrt{\cos(2y)}dy$ I don't know whether $\sin(x)...
1
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0answers
29 views

Finding functions that give nice solutions to a recurrence relation.

In a recent problem I was working through, I came across the following recurrence relation: $$ \text{K}_1(x,\ t;\ g) = 1,\quad x, t\in\Bbb{R}\quad g\in\text{C}^1 \\ \text{K}_{n+1}(x,\ t;\ g) = g'(t)\...
3
votes
2answers
79 views

Problem concerning the sequence $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$

The question is : Prove that the sequence $\{s_n\}$ where $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$ is convergent. Hence find $\lim_{n \to \infty} \left(1 - 1/2 + 1/3 - ... - 1/2n\right)$. I have ...
3
votes
1answer
49 views

A limit using the Euler number: $\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$

What is answer of this limit and how can I get it? $c$ and $i$ are constants. $$\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$$ I guess it will envolve some Neper/the Euler ...
0
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0answers
19 views

What is the method for solving this recurrence relation?

I have an equation for generating square-triangle numbers using a recurrence relation: $$f(n)^2+f(n)(2-34f(n-1))+(f(n-1)^2-70f(n-1)+1) = 0$$ But I wish to solve the equation to produce a closed form ...
2
votes
1answer
78 views

Help with $\int_0^\infty x^me^{-ax^n}dx$

I need the solution of the following integral $$\int_0^\infty x^me^{-ax^n}dx$$ where $a,n,m$ are all positive constants with $n\geq 2$. I have searched for it in the Gradshteyn but was unable to find ...
5
votes
2answers
110 views

A closed form of the series $ \sum_{n=1}^{\infty} q^n \sin(n\alpha) $

I am having problems with the following series: $$ \sum_{n=1}^{\infty} q^n \sin(n\alpha), \quad|q| < 1. $$ No restrictions on $\alpha$. I need to find out whether it converges and if yes, ...
1
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5answers
69 views

Closed form of function $f(n) = (1/n) \sum _{x=1}^{n-1} f(x)$ [closed]

Could anyone help me get to the closed form of the function: $$f(n) = \frac 1 n \sum _{x = 1}^{n-1}f(x)$$ $$f(1) = 1$$
0
votes
0answers
33 views

How can I solve this recurrence relation for generating triangle-squares?

$$N_k = 17N_{k-1} + 6(8N^2_{k-1} + N_{k-1})^{1/2} + 1$$ $$k\geqslant 1$$ I'm trying to convert a recurrence relationship for producing triangle square-numbers into a closed-form expression in terms of ...
3
votes
4answers
112 views

Evaluating series of zeta values like $\sum_{k=1}^{\infty} \frac{\zeta(2k)}{k16^{k}}=\ln(\pi)-\frac{3}{2}\ln(2) $

Somehow I derived these values a few years ago but I forgot how. It cannot be very hard (certainly doesn't require "advanced" knowledge) but I just don't know where to start. Here are the sums: $$ \...
1
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1answer
39 views

Finding a limit of a two variable function: $f(x,y)=\frac {\sin(x^2-xy)}{\vert x\vert} $

I have this exercise but not sure if I'm doing it right $$\lim_{(x,y)\to (0,0)} \frac {\sin(x^2-xy)}{\vert x\vert} $$ I assume $\frac {\sin(x^2-xy)}{\vert x\vert}\le\frac {1}{\vert x \vert} $ then ...
2
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1answer
95 views

Can you help me to evaluate $\int_{-\infty}^\infty\frac{x^2}{-1+\cosh (2x)}dx$ as $\pi^2/6$? And do you find a similar integral for $\zeta(4)$?

I was inspired in the shape of the integrals for $\zeta(2)$ in A. Córdoba, Encounters at the interface between Number Theory and Harmonic Analysis, Proceedings of the Segundas Jornadas de Teoría de ...
2
votes
0answers
74 views

Closed formula for Sum

How can I get a closed formula for this expression? $$n^{n-1}\left(1+\sum_{i=2}^{n} {\frac{2^{i-1}\cdot n!}{i^{i-1}}}\right)$$ I tried to split the sum into $$\sum_{i=2}^{n} n!=(n-1)\cdot n!$$ and ...
0
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2answers
213 views

Evaluating $\int_0^\infty\frac{\log^{10} x}{1 +x^3}dx$

How one would evaluate the following integral? $$\int_{0}^{\infty}\frac{\log^{10}(x)}{1+x^3} \, \mathrm{d}x$$ I have tried substitution with no success as well as differentiation under integral ...
0
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2answers
101 views

Find the integer part of the sum $S=\sum_{k=1}^{80} \frac{1}{\sqrt k} $

Let $$S=\sum_{k=1}^{80} \frac{1}{\sqrt k}.$$Then I would like to obtain $\lfloor S \rfloor$, the integer part of $S$. I am not able to think how to start question .
8
votes
2answers
71 views

Sum to closed form

I need to evaluate the following summation: $$ \sum_{n\in\mathbb{Z}} \frac{-1}{i(2n+1)\pi -\mu} $$ where $n$ is summed over all the integers from $-\infty$ to $\infty$ including 0. Putting this into ...
9
votes
1answer
113 views

How do we prove that $\int_{0}^{1}\int_{0}^{1}{\left(\ln{x}\ln{y}\right)^s\over 1-xy}dxdy=\Gamma^2(1+s)\zeta(2+2s)$?

How do we prove that $$\int_{0}^{1}\int_{0}^{1}{\left(\ln{x}\ln{y}\right)^s\over 1-xy}dxdy=\Gamma^2(1+s)\zeta(2+2s)$$ Integrate with respect to x first, let $s=1$ $$\int_{0}^{1}{\ln{y}\ln{x}\...
1
vote
1answer
84 views

Evaluating $\int_1^{\infty}x\: \text{erfc}(a+b \log (x)) \, dx$

I am trying to evaluate the following integral $$I = \int_1^{\infty } x \mathop{erfc}(a + b \log (x)) \, dx$$ where $a$, $b$ are some positive constants. Using the substitution $t = \log (x)$, ...
0
votes
1answer
31 views

Does there exist a closed-form expression for the following function?

I would like to find a closed-form expression for the function that is defined as follows: $T_{s}(x) = x^{s}(1 - x^{s}), \text{for prime } x \\ T_{s}(x) = x^{s}, \text{otherwise}$
4
votes
1answer
97 views

What is $\sum_{i=1}^{n}\frac{F_i}{i}$?

Mathematica is able to evaluate the summation $\sum_{i=1}^{n}\frac{F_i}{i}$ in terms of the Lerch transcendent. It is natural to consider whether or not this summation can be expressed in a more ...
0
votes
0answers
79 views

Closed-from for the series: $\sum_{k=0}^{\infty} \frac{1}{(k!)!}$

As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)!}$$ Here I don't mean the double factorial (treated here) ...
1
vote
1answer
62 views

Is there a closed-form for $\sum_{k=0}^{\infty} \frac{1}{(k!)^2}$?

As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)^2}$$ Trying on WolframAlpha, I get the value $2....
0
votes
3answers
46 views

Finding a line integral of $F(x,y) = (3x^2\cos y + 2\cos x, -x^3\sin y)$ along a given curve.

Let $F$ : $R^2 \to R^2$ be the vector field F(x,y) = ($3x^2cosy+2cosx,-x^3siny$) and $\gamma$ : [$0$,$\pi$]$\to$$R^2$ be the curve $\gamma(t)=(t,(\pi-t)^2)$. Find the line integral of $F$ along $γ$ ? ...
-2
votes
1answer
41 views

Finding a Riemann sum for $f(x)=\frac{x^2}4+2$ over $[0,2]$. [closed]

Consider the function f(x)= (x^2/4)+2. Calculate Rn for f(x)= (x^2/4)+2 on the interval [0,2] and write your answer as a function of n without any summation signs. Rn= ??? lim{n->infty} Rn= ??? i ...
4
votes
2answers
180 views

Proving the closed form $\int_{\pi/20}^{3\pi/20} \ln \tan x\,\,dx= - \frac{2G}{5}$

Context: This question asks to calculate a definite integral which turns out to be equal to $\displaystyle 4 \, \text{Ti}_2\left( \tan \frac{3\pi}{20} \right) - 4 \, \text{Ti}_2\left( \tan \frac{\pi}{...
3
votes
5answers
110 views

Evaluating $\int_ 1^2\frac{1}{x}dx$ with a Riemann Sum

I'm trying to solve $$ \int_ 1^2 \frac{1}{x} \ dx $$ using Riemann sums, however I'm having trouble solving it.