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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Closed form for the summation $\sum_{k=1}^n\dfrac{1}{r^{k^2}}$

Is there any closed form for the finite sum $$\sum_{k=1}^n\dfrac{1}{r^{k^2}}$$ or infinite sum ( when $|r|<1$) $$\sum_{k=1}^\infty\dfrac{1}{r^{k^2}} ?$$ While solving this problem, I found this ...
2
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1answer
57 views

closed form for a double sum

How can I prove that $$\underset{k\geq1}{\sum}\left(\underset{m=-\infty}{\overset{\infty}{\sum}}\frac{\left(-1\right)^{m}}{\left(2k-1\right)^{2}+m^{2}}\right)=\frac{\pi\log\left(2\right)}{8}\,?$$I ...
0
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1answer
51 views

Closed form of $\cot x=x$

I plotted the graphs of $y=\cot x$ and $y=x$. Its clear that they have infinite intersections. I tried to solve for the first root but it doesn't seem to be any known number to me. Even Wolfram Alpha ...
7
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2answers
73 views

show that $\int_0^{\infty}\sin(u\cosh x)\sin(u\sinh x)\frac{dx}{\sinh x}=\frac{\pi }{2}\sin u$

$$I(a)=\int_0^{\infty}\sin(u\cosh x)\sin(u\sinh x)\frac{dx}{\sinh x}:a>0$$ I started with $$\sin(a)\sin(b)=\frac{1}{2}(\cos(a-b)-\cos(a+b))$$ so $$I(a)=\frac{1}{2}\int_0^{\infty}\left ( ...
4
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2answers
109 views

Closed form for $ \prod_{k=1}^n (a+k^2) $

I have come across the following product: $$ \prod_{k=1}^n (a+k^2) $$ where $a$ is a positive constant. Could anyone suggest a closed form for this product? I need to approximate this for large $n$, ...
10
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181 views
+150

Does there exist a closed form for $L_k$ for any $k>3$?

I defined a sequence $L_k$ as the limit of a sequence of "hyperharmonic" series in this question. I was surprised to find that $L_3=(\sqrt{13+4\sqrt2}-1)/2$, but was unable to find a representation ...
3
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1answer
60 views

Evaluting sum $\sum_{n=0}^\infty\frac{n^k}{n!}$

Inspired by this question,I was interested if the following sum has a closed form.Looking for $k$ integer I found the Dobinski's formula so that the sum when $k$ is natural number is $e\cdot B_k$ ...
4
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3answers
45 views

Hypergeometric 2F1 with negative c

I've got this hypergeometric series $_2F_1 \left[ \begin{array}{ll} a &-n \\ -a-n+1 & \end{array} ; 1\right]$ where $a,n>0$ and $a,n\in \mathbb{N}$ The problem is that $-a-n+1$ is ...
3
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1answer
45 views

Variation on Stokes Theorem for Manifolds (2)

Let $\omega \in \Omega^0(\mathbb{R}^{2}\setminus\{0\})$ be a $0$-form such that $d\omega=0$. Is the following statement true: For any compact, oriented, $0$-dimensional submanifold $M$ of ...
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2answers
57 views

Variation on Stokes Theorem for Manifolds

Let $n >1$ and $\omega \in \Omega^{n-1}(\mathbb{R}^{n+1}\setminus\{0\})$ such that $d\omega = 0$. Is the following statement true: For any compact, oriented, $(n-1)$-dimensional submanifold $M$ ...
3
votes
3answers
118 views

Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$

For the past week, I've been mulling over this Math.SE question. The question was just to prove convergence of an infinite sum, but amazingly WolframAlpha told me it had a remarkably simple closed ...
9
votes
3answers
188 views

Closed form for ${\large\int}_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$

I'm interested in a closed form for this simple looking integral: $$I=\int_0^\infty\frac{x-\sin x}{\left(e^x-1\right)x^2}\,dx$$ Numerically, ...
5
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1answer
107 views

What is the expected number of questions answered to complete a sequence in which wrong answers send you to the start?

Given a sequence of n questions that each contain x answer choices, what is the expected number of questions answered before answering all questions correctly if answering a question incorrectly sends ...
5
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2answers
97 views

How to evaluate $\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi }{48}\ln 2$?

What is the best way to calculate the following sum?$$S=\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi^2}{48}\ln 2$$ I tried putting $$f(z) = ...
3
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1answer
102 views

How can I evaluate $\int_0^{\pi/2}\frac{x\cos{x}}{3\sin^2x+1}dx$ and $\int_0^{\pi/2}\frac{x\cos{x}}{\sin^2x+3}dx$?

I do not find the closed form of the following integrals$$\int_0^{\pi/2}\frac{x\cos{x}}{3\sin^2x+1}\mathrm dx$$ $$\int_0^{\pi/2}\frac{x\cos{x}}{\sin^2x+3}\mathrm dx$$ On the other side, I find ...
2
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0answers
45 views

What techniques does Mathematica use to find solutions to these sequences?

This question is related to my previous question: Need help finding a closed form for complicated sum. An answer to that question led my to try and find the general term of the following recurrence: ...
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2answers
124 views

Finding the closed form for $\sum_{n=1}^{\infty }\frac{\zeta (4n)}{\beta^{4n-1}}$ [closed]

Finding the closed-form $$\sum_{n=1}^{\infty }\frac{\zeta (4n)}{\beta^{4n-1}}$$ for $\beta\in(1,+\infty)$. I learned from this site many many important things but I till need more, so I need ...
1
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1answer
125 views

Need help finding a closed form for complicated sum

I'm trying to find a closed form expression for the following sequence: $$a_n=\sum_{i=1}^{n}\frac{(n-1-i+d)!}{(n-2i)!(i)!}=\sum_{i=1}^{\frac{n}{2}}\frac{(n-1-i+d)!}{(n-2i)!(i)!}$$ Where $n$ and $d$ ...
0
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0answers
26 views

Is there a closed form polynomial for this integral recursion?

While working on some statistical problems, I startet playing with integral recursions of the type $$p_{n+1}(x)=\int_a ^x \mathrm{d} y\; q(y)p_n(xy)$$ Here $q(y)$ and $p_0(x)$ are given polynomials, ...
6
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1answer
106 views

Evaluating $\int \arccos\bigl(\frac{\cos (x)}{r}\bigr)\sin^2(x){\mathrm dx}$

Following from the previous question Evaluating $\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$ I now need the extra $\sin^2x$ as in the title. Of course one power of $\sin(x)$ is easy, ...
6
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2answers
135 views

Computing a nasty integral (probably with computer algebra system)

I'm trying to do this integral, not sure if it is possible: $$ \int_{1}^{\infty}\int_{0}^{\infty} \exp\left(\, -\,{x^{2} \over y^{2}}\,\right) \exp\left(\,-\,{y^{2} \over z^{2}}\,\right) \exp\left(\, ...
1
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1answer
125 views

How to calculate $ \int_0^1 \frac{(1+x)^{2r-1}}{1+x^2}dx $?

Is there a way to express $$ \int_{0}^{1}{\left(\, 1 + x\,\right)^{2r\ -\ 1} \over \!\!\!\!\!\!\! 1 + x^{2}} \,{\rm d}x $$ in a closed form with $r\in\mathbb{N}$?
5
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0answers
82 views

Infinite series involving factorials of squares

Does $$\sum_{n=0}^\infty \frac{1}{(n^2)!}=2.04167\dots$$ possess a closed form?
3
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4answers
118 views

Computing $\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$

I want to compute $\int_0^\infty u^{-1}(1-e^{\frac{-u^2 t}{2}})\sin(u(|x|-r))\,du$ and so ,as shown below, I want to compute $$\int_0^\infty \frac{\sin(u)}{u}e^{-u^2 b} \, du$$ Attempt We split ...
4
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1answer
71 views

Sum $S=\sum _{k=1}^{\infty } \frac{(-1)^k H_k}{k^3}?$ [duplicate]

We know that $$\sum _{k=1}^{\infty } \frac{H_k}{k^3} = \frac{\pi^4}{72}.$$ Is there a closed form for the sum $$S=\sum _{k=1}^{\infty } \frac{(-1)^k H_k}{k^3}?$$ Mathematica doesn't give anything ...
4
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1answer
46 views

What's the closed-form of the Gaussian-like integral?

I once found that the integral below $$ \,{\rm I}\left(\,\alpha\,\right) =\int_{-\infty}^{\infty}\,{\rm e}^{-\left(\,x^{2}\,\, +\ \alpha\,x^{4}\,\right)} \,\,\,{\rm d}x\,,\qquad \left(\,\alpha > ...
3
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1answer
118 views

Non-additive-subtractive prime sequence

Call the following a NON additive-subtractive prime sequence or lets name it Gary's sequence. It goes like this: let a(0)=2. The next term is defined as smallest prime number which cannot be expressed ...
4
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2answers
82 views

Closed form for integral of inverse hyperbolic function in terms of ${_4F_3}$

While attempting to evaluate the integral $\int_{0}^{\frac{\pi}{2}}\sinh^{-1}{\left(\sqrt{\sin{x}}\right)}\,\mathrm{d}x$, I stumbled upon the following representation for a related integral in terms ...
0
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0answers
33 views

Closed form equation with binomial coefficients

I need a closed form for the sum $\sum\limits_{i=0}^{\infty}{n-iT-1 \choose i}x^i$ $n$, $T$ are constants and positive but may not be integers. However, they can take nearest integer values, if not ...
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0answers
56 views

Is there a closed form of $a_n=\left(\prod_{k=1}^{n-1}(4^{k}-2^{k})\right)$

In an answer to this I ended up with: $$a_n=\left(\prod_{k=1}^{n-1}(4^{k}-2^{k})\right)$$ Actually it seems this is well acceptable answer, but in any case if there exists a closed form it would be ...
1
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0answers
52 views

Closed form for generating function of Riemann Xi function

What is the closed form for $$f(x)=\ \sum_{k=1}^\infty \frac{\xi(k)x^k}{k!}$$ or $$g(x)=\frac12 \sum_{k=1}^\infty \frac{\xi(k+1/2)x^k}{k!}$$ or $$w(x)=\frac12 \sum_{k=1}^\infty ...
2
votes
0answers
75 views

Evaluating a sum $-\zeta'(2)$

Is it possible to obtain any closed-form expression for the infinite sum $$\sum_{n=1}^{\infty}\frac{\log(n)}{n^{2}}$$ by Residue calculus? My thought was to try to integrate $$f(z) ...
6
votes
2answers
99 views

Evaluating $\int_0^\infty \sqrt{\frac{x}{e^x-1}}dx$ in terms of special functions

Introduction: I've been studying integrals of the form $$\int_0^\infty \frac{x^a}{(e^x-1)^b}dx$$ where a and b are real parameters. I've been able to find closed forms for the integral in terms of the ...
4
votes
3answers
153 views

Closed form for series $\sum_{m=1}^{N}m^n\binom{N}{m}$ [duplicate]

How can we calculate the series $$ I_N(n)=\sum_{m=1}^{N}m^n\binom{N}{m}? $$ with $n,N$ are integers. The first three ones are $$ I_N(1)=N2^{N-1}; I_N(2)=N(N+1)2^{N-2}; I_N(3)=N^2(N+3)2^{N-3} $$
18
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4answers
501 views

Integrating $\int_0^\pi \frac{x\cos x}{1+\sin^2 x}dx$ [duplicate]

I am working on $\displaystyle\int_0^\pi \frac{x\cos x}{1+\sin^2 x}\,dx$ First: I use integrating by part then get $$ x\arctan(\sin x)\Big|_0^\pi-\int_0^\pi \arctan(\sin x)\,dx $$ then I have ...
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1answer
103 views

Finding the sum of this Gamma series

I am trying to compute the sum of the following series $$\sum _{k=0}^{\infty }\frac{\left(2it (1-H)^{2 (1-H)} \left(\frac{H}{\mu}\right)^{2 H} \right)^k \Gamma \left(\frac{k}{2 (1-H)}+\frac{1}{2 ...
5
votes
4answers
219 views

Integrate $I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$

I have a problem with this integral. It seems that solution has to be simple, but I couldn't find out. $$I(a) = \int_0^{\pi/2} \frac{dx}{1-a\sin x}$$ I tried using integration by parts and ...
5
votes
1answer
130 views

Evaluate $\int \ln(1 + e^x)\ \mathrm dx$

Evaluate the following indefinite integral. $$\int\ln(1 + e^x) \mathrm dx$$ My attempt :: Using integration by-parts, \begin{align} \int\ln(1 + e^x)\cdot 1\ \mathrm dx &= x\ln(1 + e^x) - \int ...
1
vote
4answers
105 views

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is ...
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Given a closed form for a series, what can be said about the sum of the squares of its terms?

Suppose I have an infinite integer sequence $\{a_k\}$, and suppose I know a closed form in terms of $n$ for this sum: $$\displaystyle\sum\limits_{k=1}^{n} a_k$$ Given this, is it always (or ever) ...
5
votes
4answers
260 views

Finding $ \int_0^1 \frac {\ln x}{1+x^2}\mathrm dx $

Today I encountered the problem of how to find $$ \displaystyle\int_{0}^{1} \frac {\ln x}{1 + x^2}\mathrm dx $$ but got no start on it. Is this one of those integrals which we have to approach from ...
8
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2answers
231 views

Closed form of $\int_{0}^{\eta}\cos nt\log\left(\frac{\cos(t/2)+\sqrt{\cos^2(t/2) -\cos^2(\eta/2)}}{\cos(\eta/2)}\right) dt$

I am reading a paper (sorry, no e-copy) with a number of infinite series, in which each term of the series is an integral of a complicated transcendental function like the one in the title. There ...
7
votes
2answers
140 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
6
votes
1answer
93 views

Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
0
votes
0answers
48 views

How to calculate the series in the modified form?

How can we calculate the series: $$ F(x)=\sum_{n=1}^{\infty}\frac{(-1)^nx^n}{1-x^n} $$ Link: how to calculate the series
3
votes
2answers
231 views

How to calculate the series?

How can we calculate the series: $$ F(x)=\sum_{n=1}^{\infty}\frac{(-1)^n}{1-x^n} $$ I found that $$ ...
8
votes
1answer
74 views

Recursively appending mean to list: Is there a closed form?

I'm pondering the following sequence: $$\begin{equation} \begin{split} a_1 & = b \\ a_{n+1} & = c\frac{1}{n}\sum_{k=1}^{n}a_k = c \times \text{mean of } \{a_1,\dots,a_n\} \end{split} ...
0
votes
1answer
45 views

How to find a recursive formula for some sequence

I know how to find a non-recursive formula for a recursively defined sequence. However, now I have this puzzle which gives me a sequence (but not the recursive definition) and challenges me to find ...
18
votes
5answers
297 views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and ...