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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2
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3answers
403 views

Generating Functions and closed form [closed]

I read somewhere that we can use generating functions to find closed form of a sequence. So what is the difference between a generating function and closed form of a sqeunce?
13
votes
5answers
482 views

Closed form for an infinite product

The following fascinating formula appears in the paper "On gamma quotients and infinite products" by M.Chamberland and A.Straub (see page 9): ...
1
vote
0answers
31 views

Calculate in closed form the number of questions to gamble to maximize the probability to get a given score in a multiple choice test

I am trying to generalize and calculate in closed form the best strategy that a gambler should follow to maximize the probability to get a given score in a multiple choice test (i.e. the number of ...
10
votes
4answers
388 views

explicit formula for recurrence relation $a_{n+1}=2a_n+\frac{1}{a_n}$

For $n\in\mathbb N$, $$a_{n+1}=2a_n+\frac{1}{a_n},\quad a_1=1. $$ Can any one give an explicit formula for all $a_n$? If such an explicit general formula doesn't exist, please explain it. I've tried ...
6
votes
3answers
114 views

Closed-form of $\int_0^1 x^n \operatorname{li}(x^m)\,dx$

I've conjectured, that for $n\geq0$ and $m\geq1$ integers $$ \int_0^1 x^n \operatorname{li}(x^m)\,dx \stackrel{?}{=} -\frac{1}{n+1}\ln\left(\frac{m+n+1}{m}\right), $$ where $\operatorname{li}$ is the ...
11
votes
1answer
95 views

Closed-forms of the integrals $\int_0^1 K(\sqrt{k})^2 \, dk$, $\int_0^1 E(\sqrt{k})^2 \, dk$ and $\int_0^1 K(\sqrt{k}) E(\sqrt{k}) \, dk$

Let denote $K$ and $E$ the complete elliptic integral of the first and second kind. The integrand $K(\sqrt{k})$ and $E(\sqrt{k})$ has a closed-form antiderivative in term of $K(\sqrt{k})$ and ...
17
votes
3answers
332 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 ...
0
votes
0answers
66 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 ...
2
votes
1answer
84 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. ...
0
votes
1answer
29 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 ...
0
votes
1answer
137 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, $$V_c = \pi r^2 H + \frac{4}{3}\pi r^3\tag1$$ ...
8
votes
2answers
242 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 ...
2
votes
0answers
123 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
2answers
263 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{?}{=} ...
6
votes
2answers
153 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 ...
2
votes
0answers
63 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}) ...
1
vote
1answer
203 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 ...
1
vote
1answer
51 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 $\dfrac{1}{2}$, ...
4
votes
2answers
127 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 ...
3
votes
1answer
56 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 ...
2
votes
1answer
55 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 ...
1
vote
2answers
133 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 ...
27
votes
2answers
1k 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 ...
3
votes
1answer
33 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
2answers
268 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 ...
11
votes
0answers
152 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\\ ...
5
votes
1answer
112 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 ...
1
vote
0answers
69 views

The integral of the product of three Meijer-G-functions

Is there any expression for the integral of the product of three Meijer-G-functions, where the domain of integration is $[0,1]$?
1
vote
2answers
105 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 ...
0
votes
1answer
71 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$$c_{k+1}=c_k+\frac{1}{(k+1)!} \ \ ; \ c_0=1$$ ...
2
votes
1answer
74 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?
4
votes
2answers
189 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 ...
2
votes
2answers
70 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) = ...
1
vote
1answer
105 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 ...
2
votes
1answer
61 views

Closed-Form Modular Arithmetic

Is there a way to define modulo division (or functions of modular arithmetic in general) as superposition of (elementary?) functions? For example, the multiplication is first introduced as ...
4
votes
1answer
113 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 ...
0
votes
3answers
140 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. ...
2
votes
0answers
56 views

I suspect this integral has a closed form but I can't find it

$$\int_a^b \frac{\text{d}\eta}{\eta}\sin\left(A\eta^3\right)\sin\left(A(\eta-B)^3\right)$$ $$\int_a^b \frac{\text{d}\eta}{\eta}\sin\left(A\eta^3\right)\cos\left(A(\eta-B)^3\right)$$ where ...
1
vote
1answer
59 views

Better closed form for generating function $\sum \binom{n}{2k} x^k$

I have a power series $F_n(x) = \sum_k \binom{n}{2k} x^k$, which has a closed form of $F_n = \frac12 \left((1 + \sqrt{x})^n + (1 - \sqrt{x})^n\right)$. $$\begin{align} (1 + \sqrt{x})^n + (1 - ...
10
votes
3answers
363 views

Evaluating the limit of a certain definite integral

Let $\displaystyle f(x)= \lim_{\epsilon \to 0} \frac{1}{\sqrt{\epsilon}}\int_0^x ze^{-(\epsilon)^{-1}\tan^2z}dz$ for $x\in[0,\infty)$. Evaluate $f(x)$ in closed form for all $x\in[0,\infty)$ ...
3
votes
2answers
91 views

Does anyone know of a closed form solution to the following integral?

Does anyone know of a closed form solution to the following integral? $$ \DeclareMathOperator\erf{erf} \newcommand{d}{\;\mathrm{d}} \int^{+\infty}_{-\infty} \erf^{\;m}\!(x) \frac{\d^n ...
4
votes
4answers
136 views

Closed form of $\sum_{n=1}^\infty (-1)^n\frac{\sin(n\theta)}{n^3}$ for $\theta\in (-\pi,\pi)$

We have to find the closed form of the following series $$\sum_{n=1}^\infty (-1)^n\frac{\sin(n\theta)}{n^3}$$ for $\theta\in (-\pi,\pi)$. We tried to use the following form of the sine ...
0
votes
1answer
49 views

Recurrence involving square root

The recurrence equation I have is: $$ T_n = c_1 + T_{n-1} + 2\sqrt{c_2 + c_1 T_{n-1}} $$ $$ T_0 = a $$ $c_1,c_2,a$ are positive real numbers I need to somehow convert this into a linear homogeneous ...
1
vote
0answers
45 views

How to solve this kind of recurrence relation in closed form? $F(n) = aF(n-1) + bF(n-2) + cF(n-3) + dF(n-4)$

How to solve this recurrence relation in closed form? $$F(n) = aF(n-1) + bF(n-2) + cF(n-3) + dF(n-4)$$ I know how to solve recurrence relations for less than four calls by solving the ...
6
votes
3answers
244 views

Closed form for a binomial series

I am wondering if any knows how to compute a closed form for the following two series. $$\sum_{m=1}^{n}\frac{(-1)^m}{m^2}\binom{2n}{n+m}$$ $$\sum_{m=1}^{n}\frac{(-1)^m}{m^4}\binom{2n}{n+m}$$ ...
0
votes
1answer
48 views

Multiplication of 2 sums that equal another multiplication of 2 sums

I have been trying to prove a formula of mine and i come across something very interesting, well to me it is. If the formula is correct, it states that: $$ \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_1 ...
1
vote
0answers
60 views

How to Evaluate this Summation to Find a Closed Form

While taking the incomplete Bell Polynomil of $x^a$ i found out that: $$ B_{n,k}^{x^a}(x) = x^{ak-n} \sum_{m=0}^k \frac{(am)!(-1)^{k-m}}{m!(k-m)!(am-n)!} $$ Now, what i am wondering is, what is the ...
27
votes
3answers
772 views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...
1
vote
1answer
33 views

Proving the closed form for an infinite sum (related to Chebyshev polynomials)

How do I prove the following identity? For $y\not= 0$, we have $$ \sum_{n=0}^{\infty} \dfrac{1}{2y}\left( (x+y)^{n+1}-(x-y)^{n+1}\right) = \dfrac{1}{(x+y-1)(x-y-1)}. $$ I am trying to find the ...
0
votes
1answer
50 views

A closed form for the coefficients of Chebyshev polynomials

The Chebyshev polynomials are defined recursively: $T_0(x)=1$; $T_1(x)=x$; $T_n(x)=2xT_{n-1}(x)-T_{n-2}(x)$ I have been trying to find a closed form for the coefficient on the monomial $x^j$ of the ...