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
votes
2answers
127 views

How to solve $\int\ x^{\ln x} dx$? [closed]

How to solve this integral $$\int\ x^{\ln x} dx$$ step by step?
11
votes
3answers
214 views

Conjecture $\sum_{m=1}^\infty\frac{y_{n+1,m}y_{n,k}}{[y_{n+1,m}-y_{n,k}]^3}\overset{?}=\frac{n+1}{8}$, where $y_{n,k}=(\text{BesselJZero[n,k]})^2$

While solving a quantum mechanics problem using perturbation theory I encountered the following sum $$ S_{0,1}=\sum_{m=1}^\infty\frac{y_{1,m}y_{0,1}}{[y_{1,m}-y_{0,1}]^3}, $$ where $y_{n,k}=\left(\...
4
votes
2answers
125 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 \sum_{n=0}^\infty\frac{(m+1)!(n+1)!}{(m+...
10
votes
2answers
691 views

Integral ${\large\int}_0^{\pi/2}\arctan^2\!\left(\frac{\sin x}{\sqrt3+\cos x}\right)dx$

I need to evaluate this integral: $$I=\int_0^{\pi/2}\arctan^2\!\left(\frac{\sin x}{\sqrt3+\cos x}\right)dx$$ Maple and Mathematica cannot evaluate it in this form. Its numeric value is $$I\approx0....
12
votes
2answers
301 views

Is there a closed form expression for the “generalized” addition of the first $n$ numbers?

Firstly, I will explain what I am trying to do intuitively. We take the sum of the first $n$ positive integers. Let's say this sum is equal to $q$. Then you add that sum to the sum of the first $q$ ...
10
votes
1answer
243 views

Need help with $\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$

Please help me to evaluate this integral: $$\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$$ Using substitution $x=2\arctan t$ it can be transformed to: $$\int_0^\infty\frac{2}{1+t^2}\...
4
votes
1answer
74 views

Is there a closed form for $n^k$ in terms of $\Delta n^{k+1},\Delta n^k$, …?

Let $\Delta$ be a sort of difference operator on a function $f(n)$ such that $$\Delta f(n)=f(n+1)-f(n)$$ Take the basic power function $f(n)=n^k$, $k\in\mathbb{N}\cup\{0\}$. Then we get $$\begin{cases}...
0
votes
1answer
28 views

What is the closed form of this series: $\sum_{n\geq 1}\frac{n^k{(-1)}^{n+1}}{n!}$ for $k<-10$ and for $k>1$?

I would like to check the closed form of this sum $$\sum_{n\geq 1}\frac{n^k{(-1)}^{n+1}}{n!}$$ , for an integer $k>1$ and $k<-10$. Note : I run some computation in wolfram alpha i have got :...
-1
votes
1answer
78 views

Closed form for $\sum_{k=1}^n\frac{1}{(2k-1)(2k+1)}$ [closed]

Find the closed form of $$\sum_{k=1}^n\frac{1}{(2k-1)(2k+1)}$$
0
votes
1answer
64 views

What is the equation of $F(x)$, given the outputs?

I can't figure out the equation of $F(x)$, if any, for the following sequence of numbers. $$225, 232, 244, 262, 287, 318, 354, 397, 446, 502, 563, 630, 704, 784, 870, 962$$ The equation should ...
4
votes
1answer
79 views

Calculating the value of $\lfloor(1+\sqrt{2})^{2n}\rfloor$

Problem: Calculate the value of $\lfloor(1+\sqrt{2})^{2n}\rfloor$ where $n$ is an arbitrary non-negative integer and $\lfloor x\rfloor$ indicates the largest integer not greater than $x$. What I ...
4
votes
2answers
94 views

closed form of $\sum_{k=0}^n {2n\choose 2k}2^k$

Is it possible to find a closed form for the expression below? $$\sum_{k=0}^n {2n\choose 2k}2^k$$ I have tried counting in two ways but made no progress. And I don't know any combinatorial ...
3
votes
1answer
77 views

Closed form of a generating function $\sum _{n=1}^\infty x^{n^2}$

I am looking for a closed form of the expression $$F(x) = \sum _{n=1}^\infty x^{n^2} $$ The question arose when I attempted to prove Lagrange's four square theorem via generating functions. It ...
10
votes
1answer
262 views

Closed form to an interesting series: $\sum_{n=1}^\infty \frac{1}{1+n^3}$

Intutitively, I feel that there is a closed form to $$\sum_{n=1}^\infty \frac{1}{1+n^3}$$ I don't know why but this sum has really proved difficult. Attempted manipulating a Mellin Transform on the ...
3
votes
1answer
68 views

Is there a closed form for the product of odd zetas?

$$\prod_{n=1}^\infty \zeta(2n+1)=\zeta(3)\zeta(5)\cdots$$ I have only managed to prove that this converges due to comparison with Euler's formula for $\zeta(2n)$ Is there a closed form for that ...
0
votes
1answer
38 views

Closed formula for sum of increasing exponents

I have a sum of the form c¹+c²+...cⁿ. Is it possible to obtain a closed formula for this, and if so how?
5
votes
2answers
103 views

Can one find a closed form solution to $\ln x=\frac{1}{x}$,

Is there a closed form solution of the equation $\ln x=\frac{1}{x}$? I couldn't find a proof myself and I don't know any theorems that says when a closed form solution exists.
4
votes
2answers
126 views

Evaluating a certain integral which generalizes the ${_3F_2}$ hypergeometric function

Euler gave the following well-known integral representations for the Gauss hypergeometric function ${_2F_1}$ and the generalized hypergeometric function ${_3F_2}$: for $0<\Re{\left(\beta\right)}<...
11
votes
3answers
421 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 $...
1
vote
1answer
45 views

For $n\geq 1$, $\sum_{k=0}^{\infty}\frac{(-1)^{k}(nk+1)^{3}}{(k+1)^6}$ in terms of $\zeta(3)$ and $\zeta(5)$ from a series calculator. Is possible?

I am doing experiments with this widget (Wolfram Alpha, a Series calculator, by HIghOPS) http://www.wolframalpha.com/widgets/view.jsp?id=86ceba9f35c96ebae137e44a36c7261a and take for Example. ...
3
votes
1answer
103 views

How to solve in radicals this family of equations for any degree $k$?

Part I. Given any constant $a,b$, the equation in $x$, $$\left(\frac{x+\sqrt{x^2+4a}}{2}\right)^{k}+\left(\frac{x-\sqrt{x^2+4a}}{2}\right)^{k}=b\tag1$$ is solvable in radicals for any degree $k$. ...
0
votes
0answers
9 views

Irregular monotonic integer sequence asymptotic to c*n

I need an example of monotonic integer sequence $a_{n}$, that would grow asympoticly to $c*n$, where $c$ is (preferably not huge) constant. I need it to behave irregularly, so it cannot be expressible ...
0
votes
0answers
18 views

Closed form solution to special cases of the algebraic ricatti equation, or ways to prove properties of the solution?

I need to solve an algebraic ricatti equation (ARE): $A'X + XA - XRX + Q = 0$ Are there special cases where I can get a closed-form for the stationary solution? If not, are there references on how ...
0
votes
2answers
35 views

Find an explicit formula for a sequence

I need to find an explicit formula for the following sequence: $$ a_0=1,\quad a_1=2,\quad a_n=2a_{n-1}+a_{n-2},\ \hbox{for}\ n\ge2 $$ I tried using the characteristic sequence and elimination method ...
2
votes
2answers
39 views

Closed formula for the power series

I have no clue how to attempt this problem. consider the power series: $$\sum_{n=0}^\infty (-1)^n \frac{x^{n+1}}{n+1}$$ Find the closed form formula for the function $f(x)$ to which the power series ...
3
votes
2answers
50 views

What is the sum of the power series: $\sum_{k=2}^\infty\frac{x^k}{k(k-1)}$?

What is the function represented by the power series $$ \sum_{k=2}^\infty\frac{x^k}{k(k-1)}\quad? $$ It looks like $\dfrac1{1-x}$ but I don't know.
19
votes
2answers
326 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
37 views

need help in finding closed form for $\sum_{i=0}^{\log(n/2)} \frac{n}{2^i}\log\frac{n}{2^i}$

I need help in finding a closed form for $$\sum_{i=0}^{\log(n/2)} \frac{n}{2^i}\log\frac{n}{2^i}$$ I am not sure even where to start. I know there is a closed form for $$f(x) = \sum_{i=0}^\infty \...
21
votes
1answer
375 views

Integral $\int_0^\infty\operatorname{arccot}(x)\,\operatorname{arccot}(2x)\,\operatorname{arccot}(5x)\,dx$

I have to evaluate this definite integral: $$Z=\int_0^\infty\operatorname{arccot}(x)\,\operatorname{arccot}(2x)\,\operatorname{arccot}(5x)\,dx$$ My CAS was only able to find its approximate numeric ...
7
votes
1answer
151 views

The elliptic integral $\frac{K'}{K}=\sqrt{2}-1$ is known in closed form?

Has anybody computed in closed form the elliptic integral of the first kind $K(k)$ when $\frac{K'}{K}=\sqrt{2}-1$? I tried to search the literature, but nothing has turned up. This page http://...
7
votes
1answer
141 views

Closed form for $\int_0^{\infty}\sin(x^n)\mathbb{d}x$

I was wondering if anyone knows a closed form for $$\mathrm{I} = \int_0^{\infty}\sin(x^n)\mathbb{d}x$$ Preliminary evaluations on Wolfram Alpha seem to yield something like this: $$\mathrm{I} = k\...
8
votes
1answer
88 views

Another polylog integral

In the interest of housekeeping, I recently took a look at what what polylogarithm integrals are still in the unanswered questions list. Some of those questions have probably languished there because ...
1
vote
2answers
68 views

Infinite product $\prod\limits_{k=0}^\infty\sum\limits_{n=0}^9z^{10^kn} $ leading to $1/(1-z)$

Please give me a hint (i am studying Complex Variables for Engineering) on how to prove that $(1+z+z^2+\cdots+z^9)(1+z^{10}+z^{20}+\cdots+z^{90})\cdots=\prod\limits_{k=0}^\infty\sum\limits_{n=0}^9z^{...
11
votes
3answers
440 views

Closed form solution to $\int_0^1\arctan^2(x)\,\sqrt{x}\,dx$

I need to compute this integral: $$\int_0^1\arctan^2(x)\,\sqrt{x}\,dx$$ I tried integration by parts, and also introducing a parameter $\arctan(a\,x)$ and differentiation wrt it, but these approaches ...
4
votes
1answer
141 views

A closed form of $\int_0^1{\dfrac{1-x}{\log x}(x+x^2+x^{2^2}+x^{2^3}+\cdots)}\:dx$ [closed]

I need some hint to calculate this integral $$\int_{0}^{1}{\dfrac{1-x}{\log x}\left(x+x^{2}+x^{2^2}+x^{2^3}+\cdots\right)}{\rm d} x$$ Regards!
1
vote
1answer
56 views

Integrate $\int_{0}^{\pi}{-\cos{x}}{_2F_1}\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\cos{^{2}x}\right)\sin{^{1+n}x}\sin{^{2}x}^{\frac{-1-n}{2}}$

May I expect the closed-form of this integral? $$\int_{0}^{\pi}{{_2F_1}\left(\left.\begin{array}{cc}\frac{1}{2}&\frac{-n+1}{2}\\&\frac{3}{2}\end{array}\right|\cos^2(x)\right)(-\cos{x})(\...
10
votes
2answers
198 views

Closed form for ${\large\int}_0^\infty\frac{\arctan(x)\,\operatorname{arccot}(x+1)}{x}dx$

I'm looking for a closed form for this integral: $$I=\int_0^\infty\frac{\arctan(x)\,\operatorname{arccot}(x+1)}{x}dx.$$ Mathematica and Maple could not evaluate it symbolically. Numerically, $$I\...
3
votes
3answers
73 views

How to evaluate $\int\sin ^3 x\cos^3 x\:dx$ without a reduction formula?

We have the integral $$\displaystyle\int \sin ^3 x \cos^3 x \:dx.$$ You can do this using the reduction formula, but I wonder if there's another (perhaps simpler) way to do this, like for example with ...
1
vote
0answers
19 views

Number of unique solutions to $\sin P_1(x, n_1) = \sin P_2(x,n_2)$

In attempting to answer this question, I was looking at the solutions for $\sin(3x - 4) = \cos(7x)$ when $0 \leq x \leq 2\pi$ (all other solutions should be multiples of these). I found $14$ distinct ...
4
votes
1answer
125 views

Closed form of the integral

$$I=\int_{-1}^1 \frac{\sin\left(\frac{\sinh x}{x}\right)\cdot\log\left(\frac{1+x}{1+x^2}\right)}{x} \space\text{d}x$$ According to Wolfram Alpha, the integral comes out to $$I=2.1607...$$ I don't ...
2
votes
3answers
143 views

Does this series have a closed-form representation?

The following sum represents the number of relevant kinds of lines in an N-dimensional tic-tac-toe game, which is why I am interested in finding a closed form, but it also is the sum of all possible ...
0
votes
3answers
42 views

Closed form expression of a summation

My prof started out with the following summation: \begin{equation} \sum_{i=0}^{k}i = \frac{k(k+1)}{2} \end{equation} Which is all fine and dandy, however the summation we want to find the closed form ...
0
votes
0answers
45 views

Finding the closed-form answer to a counting problem - polynomial result

A monic monomial of degree $m$ in $k$-many variables is considered the same as another monic monomial obtained by changing the order of the factors. For example, if $m = 4$ using variables $x, y$ and $...
1
vote
1answer
68 views

Calculating $\sum_{n=1}^\infty {\frac{nx^n}{4n^2-1}}$ [closed]

I would appreciate any help calculating the series. And determine where does the series converge uniformly. $$\sum_{n=1}^\infty {\frac{nx^n}{4n^2-1}} $$
-2
votes
2answers
50 views

Convergence and sum of $\sum_{n=0}^\infty \frac{x}{(2nx-x+1)(2nx+x+1)}$ [closed]

Find the set of $x$ where: $$\sum_{n=0}^\infty \frac{x}{(2nx-x+1)(2nx+x+1)}$$ converges and calculate the sum. Determine where does the series converge uniformly. Would appreciate any help,
2
votes
2answers
41 views

Finding a closed form solution for a recurrence

You open an account at a bank that pays 5% interest yearly, and deposit $a_0$ dollars in it. Every year you withdraw 10 times the number of years you have had the account. For example, if you started ...
2
votes
1answer
126 views

meijer g function explicit form

Can the following case of the Meijer G-function $$ G_{2,3}^{3,1}\left(z\left|\begin{smallmatrix}0,1\\ 0,0,0\end{smallmatrix}\right.\right) $$ be expressed more explicitly (in terms of other special ...
4
votes
5answers
129 views

Evaluate $\int_{\frac{-\pi}4}^{\frac{\pi}4}\ln(\sin x+\cos x)\mathrm{d}x$

$$\int_{\frac{-\pi}4}^{\frac{\pi}4} \ln(\sin x+\cos x)\mathrm{d}x $$ I just can't think of any technique to solve this question. Can anyone help me with at least how to begin?
0
votes
0answers
78 views

How Can I Find A Closed Form For This Double Summation?

I am looking for a closed form for the following summation that resembles the binomial theorem (to some degree): $$ F_n(x,z) = \sum_{k=2}^n \sum_{c=1}^{k-1} \frac{L_c^{(k-2c)}(-fg)}{(k-c)!} \left(\...
1
vote
1answer
56 views

Solving a particular nonlinear recurrence relation

I am trying to solve the recurrence relation $a_{n}=\alpha a_{n-1}^2+\beta a_{n-1}$ where $\alpha$ and $\beta$ are constants. I have been trying to find specific techniques for solving this equation ...