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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4
votes
3answers
162 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 ...
19
votes
2answers
365 views
+50

Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$

Does $~\displaystyle{\LARGE\int}_0^1\frac{\text{arctanh }x}{\tan\bigg(\dfrac\pi2~x\bigg)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$ possess a closed form expression ? This recent ...
26
votes
2answers
520 views

How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?

I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$, where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$. Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ ...
3
votes
1answer
69 views

Integral $\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$

Please help me to evaluate this integral in a closed form: $$I=\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$$ Using integration by parts I found that it could be expressed through ...
2
votes
1answer
71 views

What is $\int_0^{\pi} \frac{e^{\sin x}\cos(x)}{1+e^{\tan x}} \, dx$?

I read this question. The integral has a special property, so it might possibly be evaluable? No one tried evaluating it, so I created this. Not very often I ask question like this, but here it is. ...
1
vote
2answers
99 views

Find the value of $h$ from a Kepler-type equation

$$V = \frac{0.5r^{2}\cdot \cos^{-1}(\frac{r-h}{r})\cdot 2-\sin\big(\cos^{-1}(\frac{r-h}{r})\cdot 2\big)}{10^{6}}\tag1$$ This is the equation to find the volume of liquid in a tank in the shape of a ...
0
votes
0answers
17 views

Closed form solution (formula) for possible events

Let's have 100 time units and 4 possible events A1, A2, B1, B2 that might occur within the 100 units. A1 always occurs before A2, B1 always occurs before B2, t1 < t2 < t3 < t4. There are 2 ...
7
votes
1answer
123 views

Closed-form of $\int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

I've conjectured the following closed-form: $$ I = \int_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx = -2\,G\,\ln2, $$ where $G$ is ...
0
votes
1answer
38 views

A formula to calculate the partial volume of a capsule or tank?

We are trying to ascertain the correct formula discussed in this post. The volume formula for a capsule (a cylinder with a hemisphere at both ends) is, $$V_c = \pi r^2 H + \frac{4}{3}\pi r^3\tag1$$ ...
0
votes
0answers
12 views

Simple closed functional form for summed recurrence relation

I'm struggling to obtain a simple closed form for a summed recurrence relation. I have an overall form $y=A\left(n-\sum_i^n\frac{e^{-x_i}}{B}\right)$ where $x_{i+1}=kx_i+m$ with $A,B,m >1$ and ...
30
votes
3answers
653 views

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Is it possible to evaluate in a closed form integrals containing a squared hypergeometric function, like in this example? ...
1
vote
1answer
25 views

Summation of infinite series, where difference in consecutive denominator forms an A.P.

What is the sum of an infinite series where each term can be written as $\frac{p}{q}$, where p=1 always the difference between 2 consecutive denominators forms an A.P. For example $\dfrac{1}{2}$, ...
27
votes
6answers
902 views

Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
6
votes
3answers
152 views

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

While I was working on this question by @Vladimir Reshetnikov, I've conjectured the following closed-forms. $$ I_0(n)=\int_0^\infty \frac{1}{\left(\cosh x\right)^{1/n}} \, dx \stackrel{?}{=} ...
24
votes
4answers
491 views

Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

How to prove the following conjectured identity? $$\int_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}\stackrel{\color{#a0a0a0}?}=\frac{\sqrt[4]6}{3\sqrt\pi}\Gamma^2\big(\tfrac14\big)\tag1$$ It holds ...
1
vote
0answers
21 views

Intriguing Poisson sum with hyperbolic function

I've been playing with lots of Poisson sums lately, and I thought this one to be interesting:\ $$\sum_{k\in\mathbb{Z}}\left(\frac{1}{(k+x)\sinh{(k+x)\pi q}}-\frac{1}{\pi q (k+x)^2}\right)$$I want to ...
9
votes
3answers
413 views

Closed-form of integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy. $$ From Vladimir ...
5
votes
2answers
116 views

The value of the integral $\int_0^2\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$

The value of definite integral $$\int\limits_{0}^{2}\left(\sqrt{1+x^3}+\sqrt[3]{x^2+2x}\:\right)dx$$ is $$(A)\,4 \quad(B)\,5 \quad (C)\,6 \quad(D)\,7$$ My attempt: I tried using ...
1
vote
1answer
146 views

An Infinite series I

By decompising fractions one can show that \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1)^{2} \, (n+3)} = \frac{65}{72} - \frac{\zeta(2)}{2}. \end{align} The fraction can also be seen in the ...
6
votes
0answers
169 views
+50

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
2
votes
0answers
50 views

Is there a closed-form solution (even approximated) to this inequality?

I have the following function: $f(x, \theta) = (1-\theta)(x+1)^{-\theta}\left[ \frac{2-2\theta}{1- 2\theta} (N^{1-2\theta} - (x+1)^{1-2\theta}) - (x+1)^{-\theta}(N^{1-\theta} - (x+1)^{1-\theta}) ...
6
votes
1answer
1k views

Integral involving the error function of log(x)

Looking for a closed form for the integral $$\int_0^{\infty } e^{-\left(\frac{a-\log (x)}{b}\right)^2} \left(\frac{1}{2} \text{erf}\left(\frac{a-\log (x)}{b}\right)+\frac{1}{2}\right) \, ...
4
votes
2answers
101 views

Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$?

I hope this is not a duplicate question. If we modify the well known geometric series, with $a>1$, to $$ \sum_{n=0}^{\infty} \frac{1}{1+a^n} $$ is there a closed form with a name? I suspect ...
6
votes
3answers
211 views

Infinite Series $\sum_{k=1}^{\infty}\frac{1}{(mk^2-n)^2}$

How can we prove the following formula? $$\sum_{k=1}^{\infty}\frac{1}{(mk^2-n)^2}=\frac{-2m+\sqrt{mn}\pi\cot\left(\sqrt{\frac{n}{m}}\pi\right)+n\pi^2\csc^2\left(\sqrt{\frac{n}{m}}\pi\right)}{4mn^2}$$ ...
19
votes
2answers
330 views

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,dx$ have a closed form?

Mathematica gives an approximate result of $1.581949621806183890451628...$, but no exact form. I predict it's a function of $e$ and $\pi$, and perhaps even the Golden Ratio $\phi$ (It certainly ...
2
votes
1answer
50 views

Infinite Telescoping Sum: $\sum_{i=1}^{\infty} (X_i - X_{i-1})=$?

Let $(X_i)_{i \geq 0}$ be any countable sequence of numbers and suppose that a limit exists, so $$\lim_{i \rightarrow \infty} X_i = x.$$ Consider $\sum_{i=1}^{\infty} (X_i - X_{i-1})$. Is this ...
3
votes
1answer
51 views

Help on finding the closed form of the integral

Can anyone help me to find closed solution of the integral $$\int_0^{1-e^{-\lambda x}}\frac{u^{b-1}\,(1-u)^{a+c-1}}{[1-(1-e^{-\lambda_1 t_1})u]^{a+b+c}}\,{\rm d}u,$$ where ...
1
vote
3answers
204 views

Finding a closed-form formula for a sequence that is defined recursively

$$a_0 = 0, a_1 = 1 \quad \text{ and } \quad a_n = a_{n-1} + 2a_{n-2}\quad \text{ for }n\geq 2$$ a) Find $a_2,a_3,a_4,a_5$ b) Find a closed form-formula for $a_n$ I found the value to be ...
11
votes
0answers
121 views

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...
1
vote
0answers
50 views

Try to solve this nested radical - I have the answer [closed]

I have discovered the closed solution to the nested radical: $$\sqrt{n^{2^0-1}+\sqrt{n^{2^1-1}+\sqrt{n^{2^2-1}+\sqrt{n^{2^3-1}+\cdots}}}}.$$ I challenge anyone to find the closed form for any $n$. I ...
3
votes
1answer
30 views

What is the closed form of the following expansion

I need some help figuring out the closed form of the following expansion. T[n]=T[n-1]+T[1]*T[n-2]+T[2]*T[n-3]+T[3]*T[n-4]+...+T[n-1] I haven't done this type of ...
14
votes
1answer
262 views

Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$

Gradshteyn & Ryzhik, 7th ed., p. 570, formula 4.325(5) give the following definite integral: ...
8
votes
3answers
168 views

Sum of the series $\sum\limits_{n=0}^\infty \frac{1}{(3n+1)^3}$

The following result matches very good numerically: $$\sum_{n=0}^\infty \frac{1}{(3n+1)^3}=\frac{13}{27}\zeta(3)+\frac{2\pi^3}{81\sqrt{3}}.$$ Though I'm not sure how to approach this. How can we ...
9
votes
0answers
90 views

A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: $$\begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ ...
3
votes
1answer
96 views

How to prove $\int_{0}^{\infty}\frac{e^{-\left(\sqrt{x}-a/\sqrt{x}\right)^2}}{\sqrt{x}}dx=\sqrt{\pi},\,a>0$?

We know that $$\Gamma\left(\frac{1}{2}\right)=\int_{0}^{\infty}\frac{e^{-x}}{\sqrt{x}}dx=\sqrt{\pi} $$ but it seems that, for every $a>0 $ we have ...
1
vote
2answers
97 views

Find a closed form of the series $\sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2}$

The question I've been given is this: Using both sides of this equation: $$ \frac{x}{1-x} = \sum_{n=1}^{\infty}x^n $$ Find an expression for: $$ \sum_{n=1}^{\infty} \frac{x^n}{n^2+3n+2} $$ Any help ...
5
votes
1answer
88 views

Solve $a^x+b^x=c$ for $x$

I need to solve an equation of the form $$a^x+b^x=c$$ with $a,b\in (0,1)$ and $c\in(0,2)$ (and I'm solving for $x\in\mathbb{R}_{>0}$). I know this admits a solution (details below), but it's such ...
3
votes
1answer
31 views

A uncountable an closed subset of the Liouvilles Number

I am trying to "find" a closed and uncountable subset of the Liouville's numbers. $x\in L$ means that for all $n\in \mathbb{N}$ exists $p,q\in \mathbb{Z}$ with $q>1$ such that $$0<\vert ...
19
votes
5answers
1k views

Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$

Today, we had a math class, where we had to show, that $a_{100} > 14$ for $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$ Apart from this task, I asked myself: Is there a closed form for this ...
0
votes
1answer
64 views

Recurrence relation: $c_{k+1}=c_k+\frac{1}{(k+1)!}$

I have no idea how to proceed solving a recurrence relation like this. I know that the terms approach $e$ but beyond that I have no idea. The relation is$$c_{k+1}=c_k+\frac{1}{(k+1)!} \ \ ; \ c_0=1$$ ...
4
votes
2answers
148 views

Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$

Reading in some tables pages I found $$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$ I try to split in two sum using the roots of the ...
1
vote
1answer
450 views

Prove a non-empty subset is closed in an inner product space

I hope someone would be able to help me with the finer details of this proof. Problem: $M$ is a non-empty set in an Inner Product Space (IPS) $X$. I need to show that the annihilator of $M$ which is ...
2
votes
1answer
64 views

Solving $x - a \log(x)=b$

Let $a>0$ and $b \in \mathbb{R}$: Assume there exists an $x >0 $ s.t. $$x - a\log(x) = b$$ holds. How can it be determined in closed-form?
1
vote
1answer
97 views

Kovacic's algorithm

Is there any reference with some example, about how to solve a "riccati" equation in this (below) form :$$y'(x)+a(x)y^2(x)+b(x)y(x)+c(x)=0$$ by Kovacic's algorithm? Or can anybody help me to ...
1
vote
3answers
102 views

How do i find a closed form expression for $\sum_{k=0}^n \frac{(x-1)^k}{k+1}$?

How do I Find a closed form expression for : $$\sum_{k=0}^n \frac{(x-1)^k}{k+1}$$ Note :I have no idea how to do that, I am bad at evaluating series when we cannot use some standard series to do it. ...
38
votes
2answers
1k views

Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $

I need the method to evaluate this integral (the closed-form if possible). $$ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}\,dx $$ I used the relationship between $\tan x$ and $\tanh x$ but it didn't ...
3
votes
5answers
217 views

Infinite Series $\left(\frac12+\frac14-\frac23\right)+\left(\frac15+\frac17-\frac26\right)+\left(\frac18+\frac{1}{10}-\frac29\right)+\cdots$ [duplicate]

How do I find the sum of the following infinite series: $$\left(\frac12+\frac14-\frac23\right)+\left(\frac15+\frac17-\frac26\right)+\left(\frac18+\frac{1}{10}-\frac29\right)+\cdots$$ The series ...
3
votes
3answers
179 views

Infinite Series $1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$

Was given the following infinite sum in class as a question, while we were talking about Taylor series expansions of $\ln(1+x)$ and $\arctan(x)$: $$1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$$ ...
2
votes
2answers
67 views

HowTo solve this integral involving logarithm

I would like to solve integrals of the form $$I(c) := \int_0^\infty \log(1+x) x^{-c} \, dx ,$$ where $c \in (1,2)$. Mathematica says either 1) $I(c) = \frac{\pi}{1-c} \csc(\pi c)$ or 2) $I(c) = ...
9
votes
3answers
227 views

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

$$\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$$ I have difficulty to evaluating above integrals. First I try the substitution $x^4 =t$ or $x^4 +x^2+1 =t$ but it makes ...