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
1answer
37 views

Sum of nth powers of Fibonacci numbers

Is a closed form for $$\sum_{i=1}^n{F_i^k}$$ (where $F_i$ is the $i^{th}$ Fibonacci number and $k$ is constant) known?
0
votes
0answers
18 views

“Solving” for a sequence given an (expected) expression for the summation

Consider the "equation" \begin{equation} \frac{1}{a_n}\sum_{k=1}^n ka_k = \mathcal{O}\left(\frac{n^2}{\log n}\right).\tag{1}\label{eq:conjec} \end{equation} Does there exist some monotonically ...
6
votes
0answers
120 views

Integral ${\large\int}_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx$

Could you please help me to find closed form expressions for the following definite integrals: $$I_1=\int_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx\approx0.3606065973884796896...$$ $$I_2=\int_0^{\pi/3}...
2
votes
0answers
28 views

Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
4
votes
0answers
94 views

$\displaystyle\int_1^2\sqrt\frac{x^6+4x^4-2x^3+1}{x^4}\ \mathrm dx$ [on hold]

Find the value of: $\displaystyle\int_1^2\sqrt\frac{x^6+4x^4-2x^3+1}{x^4}\ \mathrm dx$ I do not really know where to start, so please forigve me for not showing my attempt. Wolfram alpha gives $2....
3
votes
2answers
88 views

A convergent series: $\sum_{n=0}^\infty 3^{n-1}\sin^3\left(\frac{\pi}{3^{n+1}}\right)$

I would like to find the value of: $$\sum_{n=0}^\infty 3^{n-1}\sin^3\left(\frac{\pi}{3^{n+1}}\right)$$ I could only see that the ratio of two consecutive terms is $\dfrac{1}{27\cos(2\theta)}$.
4
votes
3answers
136 views

Closed form for $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$

Is there any closed form for the infinite product $\prod_{l=1}^\infty \cos\dfrac{x}{3^l}$? I think it is convergent for any $x\in\mathbb{R}$. I think there might be one because there is a closed form ...
6
votes
3answers
136 views

Proving $\sum\limits_{k=0}^n \sum\limits_{j=0}^{n-k} \frac{(k-1)^2}{k!} \frac{(-1)^j}{j!} =1$ without character theory

Let $n \geq 2$ be an integer. I would like to prove the following identity in an easy way: $$\sum\limits_{k=0}^n \left( \frac{(k-1)^2}{k!} \sum\limits_{j=0}^{n-k} \frac{(-1)^j}{j!} \right)=1$$ You ...
3
votes
2answers
153 views

Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$

$\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$ is approximately $.5229461921333351$ but I've been assured that there is a closed form for this ...
1
vote
3answers
69 views

Average value of $f(x)=\int_x^1 \cos(t^2) dt$ on the interval $[0,1]$.

Find the average value of the function $$f(x)=\int_x^1 \cos(t^2) dt $$ on the interval $[0,1]$.
0
votes
3answers
42 views

Does this sequence have a closed form representation?

We know that $$ \sum_{s=0}^\infty \frac{\lambda^{s}}{s!} = e^\lambda$$ Relatedly, $$ \sum_{s=1}^\infty \frac{\lambda^{s}}{s!}s = \lambda \sum_{s=1}^\infty \frac{\lambda^{s-1}}{(s-1)!}$$ For which ...
1
vote
2answers
102 views

Closed form for $\int_0^1 d u \, \frac{1}{u + \lambda} \ln \left(\frac{1 + u}{1 - u} \right)$

The parameter $\lambda$ is complex and it's not on the real axis. There are some similar cases: Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$ Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx $ ...
2
votes
2answers
80 views

Calculate two sums: $\sum_{i=1}^{99}\frac{1}{\sqrt{i+1}+\sqrt{i}}$, $\sum_{i=1}^{9999}\frac{1}{(\sqrt{i}+\sqrt{i+1}) (\sqrt[4]{i}+\sqrt[4]{i+1})}$.

Calculate $$\sum_{i=1}^{99}\frac{1}{\sqrt{i+1}+\sqrt{i}}$$ I've figured out that the answer is 9 -there is a pattern that I've figured out. I've created a code and solved it... but how could I do it ...
3
votes
2answers
185 views

Evaluate $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+…}}}}}$ [duplicate]

$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+...}}}}}$ My Attempt: I tried to use the regular way. $A=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+...}}}}}$ $A^2=1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{...
11
votes
1answer
178 views

Closed form of infinite product $\prod\limits_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$

I encountered this infinite product while solving another problem: $$P(x)=\prod_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$$ $$P(x)=P \left( \frac{1}{x} \right)$$ I strongly ...
5
votes
0answers
108 views

Polynomials with degree $5$ solvable in elementary functions?

Quadratic, cubic and quartic polynomials are solvable in radicals, so there is no question here. What about the polynomials of degree $5$ (quintic)? Do we know all such polynomials (classes of ...
0
votes
0answers
51 views

Compute the partial sums in a closed-form of $\sum_{n=1}^\infty\frac{e^{-nx}}{e^{nx}-1}$, with $x>0$ or a related series

One can do the change of variable $x=nv$ in the integral formula (3) here page 2 to get after summation $$\zeta(s)^2-\zeta(s)-\frac{1}{s-1}\zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty\sum_{n=1}^\infty \...
1
vote
0answers
51 views

Does this probability statement have a closed form? (Extreme value distribution)

Problem Statement: Does this probability statement have a closed form solution? $\mathbb{P}\left(\min\left\{ w,p\right\} >c\right)=\mathbb{P}\left(\min\left\{ \left(\frac{a+\epsilon_{1}-\...
0
votes
0answers
43 views

Has it been proven that there is no closed form for the hailstone numbers?

I know none has been found, and there probably isn't one considering the effort people have put into it, but has it been proven? (for some reasonable definition of "closed form"). I'm mostly ...
5
votes
1answer
126 views

Does this integral $\int_0^\infty \frac{dx}{(1+e^x)(a+x)}$ have a closed form?

Note that $a>0$, thus I'm not sure if we can apply residues here. (For $a=0$ the integral doesn't converge). $$\int_0^\infty \frac{dx}{(1+e^x)(a+x)}$$ Despite the simple expression under the ...
3
votes
0answers
61 views

Integration of Laguerre polynomial $\int_{0}^{x}u^{p-1}(1-u)^{q-1}e^{-\theta u}L_n^{(m)}(\theta u)\mathrm du$

It's been several days that I'm confronted to this integral, without much success in its resolution. To give you more details, in my case: $n$ is an integer $>1$ $m=n-2$ $p,q \in \{n-1, n\}$ $x ...
17
votes
3answers
281 views

Integral $\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx$

I would like to evaluate this integral: $$\mathcal F(a)=\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx,\quad a>0.\tag1$$ For all $a>0$ the integrand is a smooth ...
6
votes
0answers
187 views

Juantheron-like integral

While seeing this post, the following integral is just struck me \begin{equation} \int_0^\infty \frac{dx}{(1+x^2)(1+\tan x)}\tag1 \end{equation} I have tried like what user @OlivierOloa did in ...
2
votes
2answers
70 views

Evaluate $\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$ for fixed integers $n,k\geq 1$

My question is the following Question. Can you compute some of the following $$c_{n,k}=\int_0^{\pi/2}(\sin x)^n e^{-(2+\cos x)\log k}dx$$ where $n\geq 1$ is a fixed integer and $k\geq 1$ is ...
3
votes
1answer
159 views

Contour integral for finding $\displaystyle\int_{0}^{\infty}\frac{\ln x}{(x+a)^2+b^2}dx$

I can't prove the following result: $\displaystyle\int_{0}^{\infty}\frac{\ln x}{(x+a)^2+b^2}dx=\frac{\ln \sqrt{a^2+b^2}}{b}\arctan\frac{b}{a}$ for all $a,b \in \mathbb{R}.$ Well, I consider $\...
1
vote
2answers
38 views

General Two-State Markov Chain: $P(X_{n}=1)=\frac{b}{a+b}+(1-a-b)^n \big(P(X_0=1)-\frac{b}{a+b}\big)$

Consider a general chain with the state space $S=\{1,2\}$ and write the transition probability as $$\begin{pmatrix} 1-a&a\\ b&1-b\end{pmatrix}$$ Use the Markov property to show that $$P(X_{n}=...
0
votes
1answer
27 views

Summation Closed form for floor$\left(\log_n\right)$

The closed sum for the floors of logs of consecutive integers is: $$\sum_{i=0}^n \lfloor \log_2i\rfloor = n\lfloor \log_2n\rfloor-2^{\lfloor \log_2n\rfloor+1}+\lfloor \log_2n\rfloor+2$$ This formula ...
8
votes
3answers
152 views

Prove $\int_{0}^{2\pi}{x\sin^3(x)\over 1+\cos^2(x)}dx=2\pi-\pi^2$

Integrate $$I=\int_{0}^{2\pi}{x\sin^3(x)\over 1+\cos^2(x)}dx=2\pi-\pi^2$$ $${1\over 1+y}=\sum_{n=0}^{\infty}(-1)^ny^n$$ Setting $y=\cos(x)$ $\sin^3(x)={1\over 4}{(3\sin(x)-\sin(3x))}$ ...
13
votes
1answer
136 views

Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

The Dedekind eta function is denoted by $\eta(\tau)$, and is defined on the upper half-plane ($\Im \tau >0$). Put $\tau = i x$ where $x$ is a positive real number. The function has the following ...
7
votes
5answers
190 views

Evaluation of $\sum^{\infty}_{n=0}\frac{1}{16^n}\binom{2n}{n}.$

Prove that $\displaystyle \int_{0}^{\frac{\pi}{2}}\sin^{2n}xdx = \frac{\pi}{2}\frac{1}{4^{n}}\binom{2n}{n}$ and also find value of $\displaystyle \sum^{\infty}_{n=0}\frac{1}{16^n}\binom{2n}{n}.$ $\...
2
votes
1answer
49 views

Exponential series property: $\alpha (z)=\sum_{n=-\infty }^{\infty }e^{-zn^{2}\pi}$

Good day, please a dude, show that if $$\alpha (z)=\sum_{n=-\infty }^{\infty }e^{-zn^{2}\pi}$$ then $\alpha(z^{-1})=z^{\frac{1}{2}}\alpha (z)$ for $\Re(z)>0$. I thought for properties of $e$, ...
3
votes
2answers
170 views

Prove $1-2\int_{0}^{\infty}{\tan^{-1}(x)\over e^{x\pi}-1}dx=\ln(2)$

I want to prove the following: $$J=1-2\int_{0}^{\infty}{\tan^{-1}(x)\over e^{x\pi}-1}dx=\color{red}{\ln(2)}.$$ My attempt: we know that $$\tan^{-1}(x)=x-{x^3\over 3}+{x^5\over 5}-\cdots$$ ...
0
votes
2answers
71 views

Other types of closed form for $\sum_{n=1}^\infty \frac{\cos(nx)}{n}$ in specific interval

Is there other types of closed form for the following sum? $$\sum_{n=1}^\infty \frac{\cos(nx)}{n}$$ needs to be valid over $ 0\le{x}\le{1}$ and need to be real numbers. The form below does not ...
2
votes
1answer
47 views

Solving a linear recurrence equation: $S_n=\sum_{i=0}^{n-1}a_{n-i}S_i$

I am wondering whether it is possible to express the following recurrence relation in terms of the serie $a$ and $S_{0}$. $$ S[0] = S_{0} $$ $$ \forall n > 0 \text{ }S[n] = \sum\limits_{i=0}...
1
vote
0answers
28 views

On series of the kind $\sum_{n=1}^\infty\frac{1}{n^2}\cdot\frac{1}{(1+nx)^s}$ and Frullani's theorem

I would like to know if my computations, in this post are not required justifications for the computations unless if there is a mistake in my claims, were rights and solve a question as if are known a ...
3
votes
4answers
151 views

Solve integral $\int_{-1}^{1} \frac{dx}{(e^x+1)(x^2+1)}$

Solve following integral: $$ \int_{-1}^{1} \frac{dx}{(e^x+1)(x^2+1)} $$ I tried various methods but without success.
7
votes
2answers
167 views

Integral ${\large\int}_0^\infty\big(2J_0(2x)^2+2Y_0(2x)^2-J_0(x)^2-Y_0(x)^2\big)\,dx$

I'm interested in the following definite integral: $$\int_0^\infty\big(2J_0(2x)^2-J_0(x)^2+2Y_0(2x)^2-Y_0(x)^2\big)\,dx,\tag1$$ where $J_\nu$ and $Y_\nu$ are the Bessel functions of the first and the ...
10
votes
3answers
175 views

Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$

Numerical calculations suggest that $$\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,\int_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx=1.553767373413083673460727...$$ How can we prove ...
1
vote
1answer
49 views

The best notation for this identity involving pentagonal numbers $\omega(n)$ and the $3x+1$ map

Let the $3x+1$ map $$ f(n) = \begin{cases} 3n+1 & \text {if $n$ is odd} \\ \frac{n}{2} & \text {if $n$ is even} \end{cases} .$$ Now we read the Wikipedia's page for the Collatz ...
-2
votes
2answers
61 views

Simple but hard 2 by 2 system in $x$ and $y$ [duplicate]

Is there a systematic way of solving this system, analytically? $$\begin{cases} x \ + \ y^2=11\\ x^2+y\ \ =\ 7\\ \end{cases} $$ I mean, other than brute-force.
7
votes
3answers
136 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.
13
votes
1answer
109 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 a_{n+1}=\frac{...
3
votes
1answer
128 views

Correct or not? $\int_{0}^{\frac{\pi}{2}}\frac{x^2}{x^2+\ln^2(2\sin(x))}dx\stackrel?=\frac{\pi}{8}\left[\frac{\zeta(2)}{2}+\ln(2\pi)\right]$

I got the idea from here (1) $$\int_{0}^{\frac{\pi}{2}}\frac{x^2}{x^2+\ln^2(2\sin(x))}dx=\frac{\pi}{8}\left[\frac{\zeta(2)}{2}+\ln(2\pi)\right]$$ I am not quite sure it corrects, because I check it ...
0
votes
1answer
50 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 $Y>1$. ...
7
votes
2answers
92 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, $$\prod_{n=1}^{\infty}\left(1+\frac{1}{F_{...
13
votes
0answers
105 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 a_{n+1}=\frac{a_n+b_n}2,...
5
votes
1answer
66 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 \end{...
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 ...