A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

learn more… | top users | synonyms

5
votes
3answers
209 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
1answer
38 views

Formula for number of faces in 4 dimensions

If a polytope has $m$ faces in 3 dimensions, how many faces does its analogous polytope have in four dimensions? Does a formula exist? For example, if $m=4$, you have a tetrahedron, and the 4-...
11
votes
2answers
189 views

Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$

Does a closed form exist for $$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$ in terms of $k$ and other functions? The best that I have been able to do is solve the case where $k=1$, since the ...
4
votes
4answers
81 views

Sum to infinity of trignometry inverse: $\sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right)$

If we have to find the value of the following (1) $$ \sum_{r=1}^\infty\arctan \left(\frac{4}{r^2+3} \right) $$ I know that $$ \arctan \left(\frac{4}{r^2+3} \right)=\arctan \left(\frac{r+1}2 \right)-\...
1
vote
2answers
57 views

Proportionally Distributing $N$ items across $B$ bins.

My question is similar to this: Proportional Distribution My problem follows: I have $N$ items that cannot be broken up into fractional components, but should be distributed across $B$ bins where ...
5
votes
2answers
108 views

A closed form of the series $ \sum_{n=1}^{\infty} q^n \sin(n\alpha) $

I am having problems with the following series: $$ \sum_{n=1}^{\infty} q^n \sin(n\alpha), \quad|q| < 1. $$ No restrictions on $\alpha$. I need to find out whether it converges and if yes, ...
6
votes
5answers
456 views

Trigonometry Olympiad problem: Evaluate $1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$

Find the value of $$1\sin 2^{\circ} +2\sin 4^{\circ} + 3\sin 6^{\circ}+\cdots+ 90\sin180^{\circ}$$ My attempt I converted the $\sin$ functions which have arguments greater than $90^\circ$ to $\...
15
votes
1answer
174 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
2answers
185 views

Value of this convergent series: $\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$

What is the value of- $$\frac{1}{3!}+\frac2{5!}+\frac3{7!}+\frac{4}{9!}+\cdots$$ I wrote it as general term $\sum\frac{n}{(2n+1)!}$. As the series converges it should be telescopic (my thought). But ...
5
votes
4answers
117 views
2
votes
3answers
70 views
0
votes
1answer
44 views

Path length of Gaussian

I am trying to find the path length of a Gaussian $f(x)=e^{-x^2/a^2}$ from $x=0$ to some positive point $x_0$. I've tried this by integrating the differential length, $ds^2=dx^2+dy^2$, but getting ...
0
votes
2answers
65 views

Maxwellian integral : is there a closed form?

$f_A(x,y)=\int_0^\infty du \frac{u \left(e^{-\frac{(u-x)^2}{2 A}}-e^{-\frac{(u+x)^2}{2 A}} \right)}{\sqrt{2 \pi } \sqrt{A} x \left(y^2+u^2\right)} $ is there a closed form? I was able to find ...
2
votes
1answer
42 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 ...
7
votes
2answers
230 views

Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured

By trial and error I have found numerically $$\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}=\frac{1}{6}-\frac{1}{2\pi}$$ how can this result be derived analytically?
3
votes
1answer
68 views

Another way to evaluate $\int\frac{\cos5x+\cos4x}{1-2\cos3x}{dx}$?

What I've done is this:$$\int\dfrac{\cos5x+\cos4x}{1-2\cos3x}{dx}$$ $$\int \dfrac{\sin 3x}{\sin 3x}\left[\dfrac{\cos5x+\cos4x}{1-2\cos3x}\right]{dx}$$ $$\dfrac {1}{2}\int\dfrac{\sin 8x -\sin 2x +\sin ...
29
votes
1answer
568 views

Is this integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}xdx$ equal to zero?

My initial question was to find if this integral $$ \int_0^1 \left(\left\{\frac 1x\right\}-\frac12\right)\frac{\log(x)}{x}dx$$ is convergent or divergent. ($\left\{\frac 1x\right\}$ is the fractional ...
1
vote
2answers
76 views

Is there a closed form for this binomial sum?

I am looking for a closed form of this sum:$\sum\limits_{j=k}^n\binom{j}{k}(-1)^j$ I know that this sum has a closed form: $\sum\limits_{j=k}^n\binom{j}{k}=\binom{n+1}{k+1}$ I can get this closed ...
17
votes
2answers
433 views

Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$

I was trying to find closed form generalizations of the following well known hyperbolic secant sum $$ \sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n}=\frac{\left\{\Gamma\left(\frac{1}{4}\right)\right\}^2}{...
2
votes
1answer
44 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 ...
5
votes
4answers
216 views

Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $

Let $\gamma$ be the Euler-Mascheroni constant. I'm trying to prove that $$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $$ I tried introducing a parameter to the ...
3
votes
4answers
171 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.
-2
votes
1answer
41 views

A question on the Laplace Transform of $f(t)=t e^{at}\sin (bt)$ [closed]

I would like to solve the Laplace transform of the following function: $$t \mapsto t e^{at}\sin (bt).$$ I know that $\mathscr{L}\left(\sin(bt)\right)=\dfrac{b}{s^2+b^2}$ and that you have to ...
3
votes
1answer
44 views

$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, for complex variable $z$.

I want to find this limit for complex variable $z$ $$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$$ In the real case I know $\sin(z)$ is bounded by $-1, 1,$ and the limit is $0$. But in the complex case ...
1
vote
3answers
90 views

How to prove $\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right)$?

I need help, on how to prove $$\ln{6}=\sum_{n=1}^{\infty}\sum_{r=2}^{\infty}\left({1\over r^{2n}}+{2\over (r+1)^{2n}}+{1\over (r+2)^{2n}}\right).$$ Any hints?
19
votes
4answers
379 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: $$\begin{align}\operatorname{Cl}_2(x)&=-\int_0^x\ln\left(2\sin\left(\...
8
votes
0answers
136 views

Closed form of $\int_0^1 \tan(\gamma\sqrt{1-x^2}) dx$

Some context: I'm studying the problem of nonperturbative pair creation from strong fields in quantum electrodynamics. For certain time dependent electric fields I can get some information about the ...
2
votes
3answers
68 views

Definite Integral problem: $\int_0^{\infty}\dfrac{e^{-sk}\sin (k x)}{k} \: dk$

We're given : $\int_0^{\infty}e^{-sk}\sin (k x)\:dk$ = $\dfrac{x}{x^{2}+s^{2}}$ We need to evaluate : $\int_0^{\infty}\dfrac{e^{-sk}\sin (k x)}{k} \: dk$ I tried as follows : $\int_0^{\infty}\dfrac{...
20
votes
3answers
457 views

The entry-level PhD integral: $\int_0^\infty\frac{\sin 3x\sin 4x\sin5x\cos6x}{x\sin^2 x\cosh x}\ dx$

I hope you find this integral interesting. Evaluate $$\int_0^\infty\frac{\sin\left(\,3x\,\right)\sin\left(\,4x\,\right) \sin\left(\,5x\,\right)\cos\left(\,6x\,\right)}{x\,\sin^{2}\left(\,x\,\...
1
vote
1answer
44 views

Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $

As the question says, $$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$ where a is a constant, $a>0$.
2
votes
0answers
401 views

Double sum with binomial coefficients $\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$

Find a closed form formula for this sum: $$\sum_{1\le i<j\le m} \sum_{\substack{1\le k,l\le n \\ k+l\le n}}{n\choose k} {n-k\choose l} (j-i-1)^{n-k-l}$$ It's quite likely that it can be done ...
5
votes
0answers
98 views

Summation of $\sum_{n=0}^\infty e^{-\sqrt n}$

Is there a closed form for the following sum? $$\sum_{n=0}^\infty e^{-\sqrt n}$$
3
votes
1answer
101 views

Evaluation of $\int_{0}^{1}\frac{x^{2015}-1}{\ln x}dx$

Evaluation of $\displaystyle \int_{0}^{1}\frac{x^{2015}-1}{\ln x}dx\;\;$ $\bf{My\; Try::}$ Let $$I(a) = \int_{0}^{1}\frac{x^{a}-1}{\ln x}dx\;,$$ Then $$I'(a) = \int_{0}^{1}\frac{x^a\cdot \ln(x)}{\ln(...
4
votes
4answers
125 views

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $?

How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $ ? I tried substituting $x=1/t$ but that's making it more complicated. Any suggestions?
1
vote
1answer
28 views

Closed Form Expressions: Summation and Product Operators [closed]

My question is: are expressions utilizing summation, $\Sigma$, and product, $\Pi$, operators considered 'closed-form'? To be more precise if the bounds in our summation/product operators contain ...
1
vote
1answer
55 views

Evaluating $\int_{s^{-1/n}}^{\infty}v^2\exp{\left[-\left(\frac{l}{v} + m v\right)^2\right]}dv$

I am trying to evaluate the following $$I = \int_0^s u^{-3n-1} \exp{\left[-\left(l u^n + \frac{m}{u^{n}}\right)^2\right]}\,du,$$ where $l, m$ and $n$ are positive constants. I tried to substitute $v ...
3
votes
2answers
79 views

Problem concerning the sequence $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$

The question is : Prove that the sequence $\{s_n\}$ where $s_n = 1 + 1/2 +\cdots+ 1/n - \log n$ is convergent. Hence find $\lim_{n \to \infty} \left(1 - 1/2 + 1/3 - ... - 1/2n\right)$. I have ...
14
votes
2answers
337 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 ...
2
votes
4answers
80 views

Help with the integral $\int x\sqrt{\frac{1-x^2}{1+x^2}}dx$

I would like to know what is $$\int x\sqrt{\frac{1-x^2}{1+x^2}}dx.$$ I put $x=\tan(y)$ to get integral of $\displaystyle \int \frac{\sin(y)}{\cos^3(y)}.\sqrt{\cos(2y)}dy$ I don't know whether $\sin(x)...
3
votes
1answer
48 views

A limit using the Euler number: $\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$

What is answer of this limit and how can I get it? $c$ and $i$ are constants. $$\lim_{n\rightarrow\infty}\frac{n!}{(n-i)!}\left(\frac{c}{n}\right)^{n-i}$$ I guess it will envolve some Neper/the Euler ...
1
vote
0answers
28 views

Finding functions that give nice solutions to a recurrence relation.

In a recent problem I was working through, I came across the following recurrence relation: $$ \text{K}_1(x,\ t;\ g) = 1,\quad x, t\in\Bbb{R}\quad g\in\text{C}^1 \\ \text{K}_{n+1}(x,\ t;\ g) = g'(t)\...
2
votes
1answer
76 views

Help with $\int_0^\infty x^me^{-ax^n}dx$

I need the solution of the following integral $$\int_0^\infty x^me^{-ax^n}dx$$ where $a,n,m$ are all positive constants with $n\geq 2$. I have searched for it in the Gradshteyn but was unable to find ...
0
votes
0answers
19 views

What is the method for solving this recurrence relation?

I have an equation for generating square-triangle numbers using a recurrence relation: $$f(n)^2+f(n)(2-34f(n-1))+(f(n-1)^2-70f(n-1)+1) = 0$$ But I wish to solve the equation to produce a closed form ...
0
votes
0answers
33 views

How can I solve this recurrence relation for generating triangle-squares?

$$N_k = 17N_{k-1} + 6(8N^2_{k-1} + N_{k-1})^{1/2} + 1$$ $$k\geqslant 1$$ I'm trying to convert a recurrence relationship for producing triangle square-numbers into a closed-form expression in terms of ...
1
vote
5answers
69 views

Closed form of function $f(n) = (1/n) \sum _{x=1}^{n-1} f(x)$ [closed]

Could anyone help me get to the closed form of the function: $$f(n) = \frac 1 n \sum _{x = 1}^{n-1}f(x)$$ $$f(1) = 1$$
3
votes
4answers
111 views

Evaluating series of zeta values like $\sum_{k=1}^{\infty} \frac{\zeta(2k)}{k16^{k}}=\ln(\pi)-\frac{3}{2}\ln(2) $

Somehow I derived these values a few years ago but I forgot how. It cannot be very hard (certainly doesn't require "advanced" knowledge) but I just don't know where to start. Here are the sums: $$ \...
1
vote
1answer
84 views

Evaluating $\int_1^{\infty}x\: \text{erfc}(a+b \log (x)) \, dx$

I am trying to evaluate the following integral $$I = \int_1^{\infty } x \mathop{erfc}(a + b \log (x)) \, dx$$ where $a$, $b$ are some positive constants. Using the substitution $t = \log (x)$, ...
0
votes
2answers
101 views

Find the integer part of the sum $S=\sum_{k=1}^{80} \frac{1}{\sqrt k} $

Let $$S=\sum_{k=1}^{80} \frac{1}{\sqrt k}.$$Then I would like to obtain $\lfloor S \rfloor$, the integer part of $S$. I am not able to think how to start question .
0
votes
2answers
212 views

Evaluating $\int_0^\infty\frac{\log^{10} x}{1 +x^3}dx$

How one would evaluate the following integral? $$\int_{0}^{\infty}\frac{\log^{10}(x)}{1+x^3} \, \mathrm{d}x$$ I have tried substitution with no success as well as differentiation under integral ...
1
vote
1answer
39 views

Finding a limit of a two variable function: $f(x,y)=\frac {\sin(x^2-xy)}{\vert x\vert} $

I have this exercise but not sure if I'm doing it right $$\lim_{(x,y)\to (0,0)} \frac {\sin(x^2-xy)}{\vert x\vert} $$ I assume $\frac {\sin(x^2-xy)}{\vert x\vert}\le\frac {1}{\vert x \vert} $ then ...