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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10
votes
1answer
246 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
67 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
34 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
116 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 ...
11
votes
3answers
367 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
44 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. ...
2
votes
1answer
85 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
8 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
17 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
33 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
38 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
49 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.
18
votes
3answers
317 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
363 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
150 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 ...
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} = ...
8
votes
1answer
82 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
62 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 ...
11
votes
3answers
395 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!
10
votes
2answers
180 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, ...
3
votes
3answers
72 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
124 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
41 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
67 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
48 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
40 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
106 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 ...
2
votes
2answers
73 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
77 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)!} ...
0
votes
1answer
57 views

Find $\lim\limits_{x\to0}\left(\frac{1}{\tan(x)}-\frac{1}{{e^x-1}}\right)$ [closed]

Please, help to find this limit. $$\lim\limits_{x\to0}\left(\frac{1}{\tan(x)}-\frac{1}{{e^x-1}}\right)$$
1
vote
1answer
53 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 ...
6
votes
4answers
232 views

A limit related to super-root (tetration inverse).

Recall that tetration ${^n}x$ for $n\in\mathbb N$ is defined recursively: ${^1}x=x,\,{^{n+1}}x=x^{({^n}x)}$. Its inverse function with respect to $x$ is called super-root and denoted $\sqrt[n]y_s$ ...
4
votes
1answer
134 views

Is there a proof that the Harmonic numbers are not an elementary function? [duplicate]

The Harmonic numbers $H_x = \sum_{n=1}^x 1/n$ are the sum of the reciprocals of the natural numbers up to a given number. The first few are $0, 1, 3/2, 11/6, \ldots$. $H_x$ can be defined for ...
3
votes
2answers
85 views

Proving the closed form of $\sin48^\circ$

According to WA$$\sin48^\circ=\frac{1}{4}\sqrt{7-\sqrt5+\sqrt{6(5-\sqrt5)}}$$ What would I need to do in order to manually prove that this is true? I suspect the use of limits, but I don't know where ...
1
vote
0answers
161 views

No closed form for $\sum_{n\in P} \frac{1}{n^2}$

I think that I can say with a fair amount of assurance that $$\sum_{n\in \mathcal P} \frac{1}{n^2}$$ has no closed form (assuming that $\mathcal P$ represents the full set of primes) I currently know ...
7
votes
1answer
235 views

How to evaluate $\int_0^{\pi /2}\frac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos u)})^2}du$?

I want to find the value of $$\int_0^{\pi /2}\dfrac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos u)})^2}du.$$ Let $v=\frac{\pi}{2}-u$, then $$\int_0^{\pi /2}\dfrac{u^2\ln{(2\cos u)}}{(u^2+\ln^2{(2\cos ...
0
votes
3answers
41 views

Closed form solution for the recurrence

I am given the following recurrence and need to find a closed form solution for the recurrence. I have no idea on how to get started though and i need some help on leading me to solve this. $A_0=20, ...
0
votes
1answer
73 views

Can I have some assistance with this integral calculation?

This integral has bothered me for the longest time: $$J=\int_{-1}^0 \sqrt[x]{2+\Gamma(x+1)}\space\text{dx}$$ This guy is extremely minuscule in relation to most other integrals but was amazingly ...
22
votes
0answers
425 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 ...
1
vote
2answers
42 views

Are there methods to recursively calculate the decimal expansion of real numbers?

Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence, $$(1) \quad C_0=0.1, \ C_1=0.101, \ ...
1
vote
0answers
34 views

What formulas are available to find the nth digit of a number?

Imagine that'd I'd like to investigate the digits of $\sqrt{2}$, or of any real number. If I want a formula for the nth digit of a real number $x$, we have, $$(1) \quad \operatorname{d_n}(x)=\lfloor ...
1
vote
1answer
42 views

Closed form for the first local min $\gt 0$ of $x!$ (in reference to x-value of min)

The first local min of $x!$ is the point $(0.461632...,0.885603...)$ Is there a close form of $0.461632...$, the $x$-value of the above point? If you can tell me the closed form, could you help me ...
6
votes
1answer
85 views

Game in a circle

$N$ players play a game. They stand in a way such that they form a regular $N$-gon. Players are numbered from $1$ to $N$. The players throw boomerangs in clockwise order, in turns. At first player $1$ ...