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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5
votes
0answers
64 views

Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{n}\left(e^{-\pi x}\right)}{1+x^2} dx $

The Jacobi theta function $\theta_4$ is defined by $$\displaystyle \theta_4(q)=\sum_{n \in \mathbb{Z}} (-1)^n q^{n^2} \tag{1}$$ For this question, set $q=\large e^{-\pi x}$ and $\theta_4 \equiv ...
6
votes
2answers
75 views

Showing that $\prod_{n=1}^{\infty}\left(1+\frac{1}{F_{2^n+1}L_{2^n+1}}\right)=\frac{3}{\phi^2}$

Infinite product $F_{n}:=[1,1,2,3,5,8,\cdots]$ and $L_{n}:=[1,3,4,7,\cdots]$ for $n=1,2,3,\cdots$ respectively. $\frac{1+\sqrt5}{2}=\phi$ Show that, ...
6
votes
3answers
113 views

Integrate $\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$

I can't solve the integral $$\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$$ I tried it by using Beta and Gamma function and integration by parts. Please help me to solve it.
11
votes
1answer
89 views

Integral with arithmetic-geometric mean ${\large\int}_0^1\frac{x^z}{\operatorname{agm}(1,\,x)}dx$

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of positive numbers $a$ and $b$ is denoted $\operatorname{agm}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad ...
18
votes
4answers
679 views

How to evaluate $I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$

Prima facie, this integral seems easy to calculate,but alas, this not's case $$I=\int_0^{\pi/2}\frac{x\log{\sin{(x)}}}{\sin(x)}\,dx$$ The numerical value is I=-1.122690024730644497584272... How to ...
37
votes
3answers
915 views

Find the value of $\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$

I need to find a closed-form for the following integral. Please give me some ideas how to approach it: $$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
0
votes
1answer
42 views

Closed form of a series

I am looking for a closed form of the following convergent series: $$\sum_{n=0}^\infty \frac{(-\lambda^2)^n}{(6n+i)!}$$ For the case of $i=0$, the answer is ready, but when $i=1,2,3,4,5$, everything ...
0
votes
0answers
16 views

Is there any closed form of an upper-bound of the following equation?

Could you please let me know if you can find the closed form of the following Equation (or any upper-bound that converges): $\sum_{i=1}^\infty(\dfrac{X}{Y^i})^i i!$, where $0<X<1$ and $Y>1$. ...
13
votes
0answers
79 views

Arithmetic-geometric mean of 3 numbers

The arithmetic-geometric mean$^{[1]}$$\!^{[2]}$ of 2 numbers $a$ and $b$ is denoted $\operatorname{AGM}(a,b)$ and defined as follows: $$\text{Let}\quad a_0=a,\quad b_0=b,\quad ...
5
votes
1answer
63 views

Combinatorial proof of a certain alternating sum of binomial coefficients

The following identity appeared as a question earlier today $$\displaystyle\sum\limits_{k=0}^n (-1)^k\binom{m+1}{k}\binom{m+n-k}{n-k} = \begin{cases} 1\ \text{if}\ n=0 \\ 0\ \text{if}\ n>0 ...
2
votes
1answer
25 views

What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau $ is a constant. But is ...
2
votes
1answer
111 views

Can anyone verify $\int_{0}^{\infty}\frac{e^{-2nx}+2nx-1}{x(e^x+1)}dx=\ln{2n\choose n}$? [closed]

Central binomial coefficient from mathworld $$\frac{2^{2n+1}}{\pi}\int_{0}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx={2n\choose n}$$ Here we have $\ln{2n\choose n}$ in term of another integral, ...
4
votes
1answer
49 views

Analytic extension of $\sum_{k=1}^n\frac1k$ complex domain

The analytic extension: $$\sum_{k=1}^n\frac1k=\int_0^1\frac{x^n-1}{x-1}dx$$ I was wondering for what values of $n$ does this extend to, mainly complex values of $n$. I know it is defined for $n=0$, ...
59
votes
2answers
2k views

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. ...
1
vote
1answer
38 views

Antisymmetric asymptotic curve with only simple binary arithmetic?

I'm looking for an s-curve formula with similar properties to $Sigmoid$ or $\tan^{-1}$, but without 'expensive' unary functions or their binary generalizations (e.g. $^x\log y$). The only allowed ...
3
votes
1answer
132 views

Finding the $n^{th}$ derivative of $\frac{x^n}{(1+x)}$

Find the $n^{th}$ derivative of $\frac{x^n}{(1+x)}$ . I think we have to use Leibnitz's Formula to evaluate this, but I haven't succeeded in it as well. I have already received an answer of $\frac ...
26
votes
1answer
558 views

Curious about an empirically found continued fraction for tanh

First of all, and since this is my first question in this forum, I would like to specify that I am not a professional mathematician (but a philosophy teacher); I apologize by advance if something is ...
2
votes
1answer
39 views

Can you provide us a good approximation for $\sum_{n=1}^{\infty} \left| \log \left( 1+\frac{\mu(n)}{n^2} \right) \right|$?

Let $a_n=\frac{\mu(n)}{n^2}$, where $\mu(n)$ is the Möbius function. Since $\sum \left| a_n \right| $ is convergent by the comparison test, then a proposition from analysis ensures that ...
5
votes
1answer
175 views

Seeking closed-form solution to $\sum_{n=1}^{\infty}\frac{\log{(1+n)}}{(1+n)^{\alpha}-1}$

I'm looking for a closed-form solution to this infinite series: $$S(\alpha):=\sum_{n=1}^{\infty}\frac{\log{(1+n)}}{(1+n)^{\alpha}-1},~~~\Re(\alpha)>1.$$ My attempt All I've really been ...
3
votes
2answers
194 views

A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$

Similar to the cfracs in this post, define the two complementary continued fractions, $$x=\cfrac{-(m+1)}{km\color{blue}+\cfrac{(-1)(2m+1)} {3km\color{blue}+\cfrac{(m-1)(3m+1)}{5km\color{blue} ...
4
votes
2answers
123 views

Finding the infinite series $\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{m!\:n!}{(m+n+2)!}$

Evaluating $$\sum_{m=0}^\infty \sum_{n=0}^\infty\frac{m!n!}{(m+n+2)!}$$ involving binomial coefficients. My attempt: $$\frac{1}{(m+1)(n+1)}\sum_{m=0}^\infty ...
1
vote
2answers
6k views

Convert Recursive to Closed Formula

I got a particular sequence defined by the following recursive function: $$T_n = T_{n-1} \times 2 - T_{n-10}$$ I need help converting it to a closed form so I can calculate very large values of n ...
2
votes
1answer
35 views

Closed form for this integral (looks like Bessel)

I'm struggling to find a closed form for the following distribution (which is after all a Fourier Transform) written in integral form: $$I=\int_0^\infty\!\!\text{d}k\ \frac{ k }{\sqrt{k^2+m^2}}\sin(k ...
4
votes
0answers
82 views

Closed form for $\int_0^1\frac{x}{\ln(x+1)(x^3+3x+3)}dx$

How can I evaluate the closed form of the following integral: $$\int_0^1\frac{x}{\ln(x+1)(x^3+3x+3)}dx$$ According to Wolfram Alpha, the numerical value of this integral is close to 0.2673, but ...
0
votes
2answers
48 views

Solution to recurrence relation, as a formula involving summation operator

Here is what I am tasked with.. Find a solution to the recurrence relation: $F(0) = 2$ $F(n+1) = F(n) + 2n^2 - 1$ as a formula involving the summation operator $$\sum_{i=1}^n$$ Sorry for the ...
1
vote
3answers
31 views

Evaluate $\int_1^N \frac{-3N+6t-3}{t^3(N-t+1)^4}dt$ when $N=3$ or $N=5$

Let the Cauchy product $$(\zeta(3))^2=\sum_{n=1}^\infty c_n,$$ where $$c_n=\sum_{k=1}^n\frac{1}{k^3(n-k+1)^3},$$ and $\zeta(3)$ is the Apèry constant. Taking $f(x)=\frac{1}{x^3(N-x+1)^3}$ in Abel's ...
13
votes
4answers
593 views

Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$

How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ? Inspired by ...
3
votes
1answer
129 views

How can we see that $ \sum_{n=0}^{\infty}\frac{2^n(1-n)^3}{(n+1)(2n+1){2n \choose n}}=(\pi-1)(\pi-3) $?

I wonder will it help me so prove it if I was to decompose it into partial fractions? Mathematica approves of the identity; it is converges. can anyone help me to prove it? $$ ...
3
votes
0answers
91 views
4
votes
1answer
113 views

Evaluating the integral $\int \frac{x^2+x}{(e^x+x+1)^2}dx$

Evaluate $$\int \frac{x^2+x}{(e^x+x+1)^2}dx$$ I tried converting in the form of Quotient rule(seeing the square in the denominator), neither am I able to make the denominators' derivative in the ...
0
votes
0answers
22 views

Calculation of an integral involving the sum of a range of natural exponential functions

Does somebody know how to solve the following integral, I extremely hope I can obtain its close-form solution: \begin{equation} \int \sqrt{ \sum_{i=1}^{M}\sum_{j=1}^{M} ...
14
votes
0answers
288 views

A closed form of $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$?

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ ...
1
vote
1answer
20 views

Is there a general formula for the $n$'th variable of the solution for a lower triangular linear system of equations?

I have a countably infinite linear system of equations $Ax = b$, where $A$ is lower triangular with $-1$ at all diagonal entries, and $b = \{-1/2,0,0,...,0\}^T$. I.e the $n$'th unknown depends solely ...
1
vote
1answer
34 views

On $-\frac{\zeta'(x)}{x\zeta(x)}$ and von Mangoldt function

I believe that it is possible show the following Fact. For real $x>e$ then $$-\frac{\zeta'(\log x)}{x\zeta(\log x)}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{\log x}},$$ where $\zeta(x)$ is the ...
0
votes
1answer
39 views

The Cauchy product $\sum_{n=1}^\infty \frac{\log n}{e^n}= \left( 1-\frac{1}{e} \right)\sum_{n=1}^\infty\frac{\log n!}{e^n} $

I know that the Cauchy product is defined $$\left(\sum_{n=1}^\infty\frac{\log n}{e^n}\right)\left( \sum_{n=1}^\infty\frac{1}{e^n} \right)= \sum_{n=1}^\infty\sum_{k=1}^n\frac{\log k}{e^{k+n-k+1}},$$ ...
2
votes
3answers
87 views

Does the infinite series $\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$ converge?

I have been wondering if this infinite series converges $$\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$$ I tried to put it in wolfram alpha but it says that the ratio test is inconclusive, but when I ...
6
votes
1answer
134 views

Simplifying a certain polylogarithmic sum in two variables

This question is related to my previous question here. While tinkering around for a solution I found that the integral there can be reduced to the problem of solving the following basic logarithmic ...
0
votes
1answer
32 views

Infinite sum of Hermite polynomials with same order, but different argument

I am looking for any possible simplification of the following sum for positive reals $\alpha,\beta$ and positive integer $n$: $$ \sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha) $$ ...
2
votes
2answers
73 views

What is the sum of this series: $1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 +\cdots$?

Say I have a series like the following; $$1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 + \frac{1 \times 6 \times 11}{5 \times 10 \times 15}x^3 + \cdots.$$ How do I find the sum of this? ...
11
votes
3answers
378 views

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

It appears that $$\int_0^\infty\frac{\tanh^2(x)}{x^2}dx\stackrel{\color{gray}?}=\frac{14\,\zeta(3)}{\pi^2}.\tag1$$ (so far I have about $1000$ decimal digits to confirm that). After changing variable ...
18
votes
2answers
321 views

Why is this definite integral antisymmetric in $s\mapsto s^{-1}$?

I recently happened into the following integral identity, valid for positive $s>0$: $$\int_0^1 \log\left[x^s+(1-x)^{s}\right]\frac{dx}{x}=-\frac{\pi^2}{12}\left(s-\frac{1}{s}\right).$$ The ...
0
votes
1answer
102 views

Proving the closed form of a generating function of the sum of n lucas numbers is equal to the n+2th lucas number

1760887     I've been working on this homework problem for a while now and can't seem to solve it. Let $L_n = L_{n-1} + L_{n-2}$ for $n\ge 2$ where $L_0 = 2$ and $L_1 = 1$ $M_n = 1 + ...
2
votes
1answer
21 views

Closed form for binomial sum with absolute value

Do you know whether the following expression has a (nice) closed form or a close enough approximation? $$\frac{1}{2^n}\sum_{k=0}^{n} \binom{n}{k}|n-2k|$$ Thanks a lot :) Cheers, M.
-2
votes
1answer
42 views

Expected value of $X^{2n}$ where $X \sim N(0,1)$ [closed]

The question is: Show that if $X ∼ N(0, 1)$ has the standard normal distribution then $E[X^{2n}] = \frac{2n!}{2^{n}n!}$ Hint: compute the integral $\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} ...
-2
votes
2answers
49 views

Guess/Find a formula just given input and output. [closed]

I am looking a formula that given the three inputs, gives the output: $$(7,8,9)=7 \\ (1,3,3)=2 \\ (65,30,74)=56 \\ (9,9,7)=8 \\ (999999999, 999999998, 1000000000 )=999999998 \\ (775140200 ,616574841 ...
1
vote
1answer
39 views

Compute the Dirichlet inverse of $f(n)=\frac{1}{1+|\mu(n)|}$, where $\mu(n)$ is the Möbius function

Let for integers $n\geq 1$ the arithmetical function defined by $$f(n)=\frac{1}{1+|\mu(n)|},$$ where $\mu(n)$ is the Möbius function. Note that $f(1)=\frac{1}{2}\neq 0$, and $f(n)$ isn't ...
3
votes
1answer
39 views

Solving the recursion $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$

Solving the recursion $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$ $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$ $p_n = p \cdot p_{n-2} +p_{n-1} - p\cdot p_{n-1}$ $p_n - p_{n-1} = (-p)(p_{n-1} - p_{n-2})$ $= ...
42
votes
3answers
4k views

A strange integral having to do with the sophomore's dream:

I recently noticed that this really weird equation actually carries a closed form! $$\int_0^1 \left(\frac{x^x}{(1-x)^{1-x}}-\frac{(1-x)^{1-x}}{x^x}\right)\text{d}x=0$$ I honestly do not know how to ...
8
votes
2answers
159 views

What is the subword complexity function of this infinite word?

Let $w_{0}$ denote the finite word $01$ in the free monoid $\{ 0, 1 \}^{\ast}$, and for $i \in \mathbb{N}$ define $w_{i}$ as the word obtained by adjoining the first $\left\lfloor ...
1
vote
2answers
68 views

Find a function for the infinite sum $\sum_{n=0}^\infty \frac{n}{n+1}x^n$

I need to find a function $f(x)$ which is equal to the sum $$ \sum_{n=0}^\infty \frac{n}{n+1}x^n, $$ for every $x\in \mathbb{R}$ for which the series converge. Now, using WolframAlpha, I've found the ...