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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0
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1answer
27 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 $$\left(\sum_{n=1}^\infty\frac{\log n}{e^n}\right)\left( \sum_{n=1}^\infty\frac{1}{e^n} \right)= \sum_{n=1}^\infty\sum_{k=1}^n\frac{\log k}{e^{k+n-k+1}},$$ ...
0
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0answers
8 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 ...
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0answers
14 views

Examples of geometric function without closed analytic form [on hold]

There are some non-closed form algebraic expressions that can be easily expressed via geometric method of form construction. [While what is geometric and what is analytic closed form not intuitive in ...
1
vote
3answers
82 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
126 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
31 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
69 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
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 ...
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
95 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 + ...
3
votes
0answers
39 views

Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{10}\left(e^{-\pi x}\right)}{1+x^2} dx $

Motivation 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 ...
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
40 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
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1answer
34 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 ...
0
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0answers
20 views

On a closed-form from particular values of the Riemann zeta function and divisor functions

I am looking if I can get a closed-form for an infinite series, but I don't know for what it is possible, without finish my computations (see my Question, below). From Applications (8.1 Infinite ...
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
158 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
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2answers
66 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 ...
5
votes
1answer
122 views

Proving there don't exist $F(x), G(x)$ such that $1^{-1}+2^{-1}+3^{-1}+\cdots+n^{-1}={F(n)}/{G(n)}$

We know the following sum is a polynomial of degree $N+1$ about $n$ where $n, N\in\mathbb N$: $$S_N(n)=\sum_{k=1}^{n}k^N=1^N+2^N+3^N+\cdots+n^N.$$ Then, I got interested in the following question: ...
35
votes
4answers
4k views

Do harmonic numbers have a “closed-form” expression?

One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
5
votes
2answers
144 views

Deducing the closed form for pentagonal numbers

Consider the sequence: $0,1,5,12,22,35,51,70,92,117,145,176,\ldots$ Find both a recurrence and a closed form for this sequence. I've done some research and found out that the majority ...
7
votes
2answers
80 views

Proving $n - \frac{_{2}^{n}\textrm{C}}{2} + \frac{_{3}^{n}\textrm{C}}{3} - …= 1 + \frac{1}{2} +…+ \frac{1}{n}$ [closed]

Prove that $n - \frac{_{2}^{n}\textrm{C}}{2} + \frac{_{3}^{n}\textrm{C}}{3} - ... (-1)^{n+1}\frac{_{n}^{n}\textrm{C}}{n} = 1 + \frac{1}{2} + \frac{1}{3} +...+ \frac{1}{n}$ I am not able to prove ...
1
vote
1answer
36 views

Z transform of $\sum_{k=0}^{n}3^{k}$

My task is to calculate z transform of signal $x[n]=\sum\limits_{k=0}^{n}3^{k}$ ? By definition, $$ \begin{align} X(z) &= \sum\limits_{n=-\infty}^{n=\infty}x[n]z^{-n} \\ &= ...
3
votes
2answers
83 views

Evaluating $\int_0^{\pi/2}({x \over \sin x})^2dx$ using value of a given integral

Question If $\int_0^{\pi/2}\ln({\sin x})dx = {\pi \over 2}\ln({1\over2})$ then find the value of $\int_0^{\pi/2}({x \over \sin x})^2dx$ I'm stumped. I have no clue what to do. A hint would be ...
4
votes
6answers
89 views

Simplifying radicals inside radicals: $\sqrt{24+8\sqrt{5}}$

Simplify: $\sqrt{24+8\sqrt{5}}$ I removed the common factor 4 out of the square root to obtain $2\sqrt{6+2\sqrt{5}}$, but the answer key says it is $2+2\sqrt{5}$. Am I missing out on some general rule ...
14
votes
1answer
114 views

Are there some techniques which can be used to show that a sum “does not have a closed form”?

I am aware that there are some techniques which can be used to show that some function does not have an antiderivative expressible using elementary functions, such as Liouville's theorem. (More ...
0
votes
1answer
77 views

Is there a closed form for $\sum_{k=0}^n \frac{x^k}{k!}$? [closed]

What is the closed form of $$\sum_{k=0}^n \frac{x^k}{k!}$$ as a function of $x$ and $n$? Knowing that it converges to $e^x$ when $n\to \infty$.
3
votes
1answer
72 views

General Form for $\displaystyle \sum_{n=1}^{\infty}\frac{d\left ( kn \right )}{n^2}$

The function d(x) gives the number of divisors of x. "k" is a positive integer. In Mathematica, I think, d(x) is implemented as DivisorSigma[0,x]. If you know of such a General Form or can point me to ...
4
votes
2answers
112 views

How can I find a closed form for the summation (i^2)(-1^i+1) systematically?

In one of my homeworks I was given the following sequence $1^2-2^2+3^2-4^2+\dots (-1)^{n+1}n^2$, and I'm supposed to find a closed form formula and prove that it works. Rewriting this as a sum gives ...
2
votes
1answer
16 views

Closed Form Solution to Exponential Recursion

Is there a closed form solution to the function $f_n=2^{f_{n-1}}$ where $f_0=2$ ? For instance, the first few values of the function are 2, 4, 16, 65536.
8
votes
1answer
190 views

How to calculate $\int_0^\pi \ln(1+\sin x)\mathrm dx$

How to calculate this integral $$\int_0^\pi \ln(1+\sin x)\mathrm dx$$ I didn't find this question in the previous questions. With the help of Wolframalpha I got an answer $-\pi \ln 2+4\mathbf{G}$, ...
1
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0answers
71 views

Integral of a Gaussian times a rational function

I have been looking as crazy for a closed-form expression of integrals of the following nature $$\int_0^\infty\text{d}x\,e^{-(a+x)^2}\frac{x^m}{(x^2+b)^n}$$ where $a,b>0$ and $n$ and $m$ are ...
0
votes
1answer
30 views

Closed form of the function

i've a function $h_j(x) =1/N\sum _{k=-N/2}^{N/2}1/c_k e^{ik(x-x_j)}$ where N is even and $c_k = 1$ when $k = -N/2 +1, ..., N/2 -1$and $c_k = 2$ when $k = -N/2, N/2$ i'm unable to calculate the closed ...
5
votes
1answer
105 views

Definite Integral $\int_0^1 \left \{\frac{(-1)^{\lfloor 1/x \rfloor}}{x} \right\}\, dx$

The curly brackets mean 'FractionalPart' which, I believe, is defined as {${x}$}$=x-\lfloor x \rfloor$ where $x \in \mathbb{R}$. The conjecture I have found but can not prove is that the definite ...
28
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3answers
539 views

A closed form for a lot of integrals on the logarithm

One problem that has been bugging me all this summer is as follows: a) Calculate $$I_3=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \ln{(1-x)} \ln{(1-xy)} \ln{(1-xyz)} \,\mathrm{d}x\, \mathrm{d}y\, ...
0
votes
1answer
56 views

General Form for a series

I am struggling to put a Series in a general form and was wondering if someone here could give a hand with that. If the question is to general or not meeting the standards, I apologize in advance. ...
5
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2answers
106 views

Definite Integral $\int_0^1 \left \{\frac{1}{x^\frac{1}{6}} \right\}\, dx$

The curly brackets mean 'FractionalPart' which, I believe, is defined as {${x}$}$=x-\lfloor x \rfloor$ where $x \in \mathbb{R}$. My best approximation so far is: .182657 , however, I suspect there ...
0
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0answers
37 views

Definite integral of error function times exponential and Gaussian

I am looking for the solution of the following integral $$\int_{-\infty}^\infty\text{d}x\,\text{erf}(x)e^{-a x^2-bx}$$ where $a$ is real but $b$ is in general complex. For the case of $b$ being real ...
38
votes
3answers
811 views

Calculate the following infinite sum in a closed form $\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$?

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
1
vote
0answers
64 views

Finding a closed form for $\sum_{n=-\infty}^{n=+\infty}\frac{1}{n^{2k}+a^{2k}}$

I am trying to find a closed form for $S=\sum_{n=-\infty}^{n=+\infty}\frac{1}{n^{2k}+a^{2k}}$, $k \in \mathbb{N^{*}}$, $a>0$ I don't even bother to look for a closed form with an odd exponent, ...
1
vote
1answer
33 views

Find a recurrence expression which solution have $\sin$ or $\cos$.

I, I'm a computer science student of the first course. My teacher have told us to try to find a recurrence equation for the closed-form expression: $$f(n) = 2^n + 3^n \cos\left(\frac{n\pi}{2}\right) ...
26
votes
0answers
367 views

Closed form for $\left(1+\left(\frac{1}{2}+\left(\frac{1}{3}+\left(\frac{1}{4}+\cdots\right)^2\right)^2\right)^2\right)^2$?

Nested squares seem to be more promising than nested radicals, since they give rational approximations and in principle can be expanded into a series. These two expressions converge numerically: ...
3
votes
1answer
78 views

Closed form for Fibonacci numbers

We know the closed form for Fibonacci number as $F_n=\frac{1}{\sqrt5}\left[\left(1+\frac{\sqrt5}{2}\right)^n−\left(1−\frac{\sqrt5}{2}\right)^n\right]$ But while finding $F_n \pmod{99991}$ the closed ...
11
votes
2answers
173 views

What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?

There are well-known closed-form evaluations for integrals of the form $\int_0^1 a(x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx $ for certain algebraic functions $a(x)$. For example, an ...
1
vote
1answer
34 views

how do I prove this by induction? (recursion)

The terms are given recursively: $P_0=3$ $P_1=7$ and $P_n = 3P_{n-1}-2P_{n-2}$ for $n\ge2$ What should I assume and what step proves that $P_n=2^{n+2}-1$ is a closed form of the sequence. Suppose ...
-1
votes
2answers
55 views

How can I prove by induction that this is a closed form of the Fibonacci sequence? [duplicate]

How can I prove by induction that this is a closed form of the Fibonacci sequence? $$F_n=\frac1{\sqrt5}\left(\frac{1+\sqrt5}2\right)^{n+1}-\frac1{\sqrt5}\left(\frac{1-\sqrt5}2\right)^{n+1}$$ I've ...
8
votes
3answers
192 views

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

Wikipedia gives $$\sum_{k=2}^\infty(\zeta(k)-1)=1,\quad\sum_{k=1}^\infty(\zeta(2k)-1)=\frac34,\quad\sum_{k=1}^\infty(\zeta(4k)-1)=\frac78-\frac\pi4\left(\frac{e^{2\pi}+1}{e^{2\pi}-1}\right)$$ from ...
2
votes
3answers
80 views

Sum of a series $\frac {1}{n^2 - m^2}$ m and n odd, $m \ne n$

I was working on a physics problem, where I encountered the following summation problem: $$ \sum_{m = 1}^\infty \frac{1}{n^2 - m^2}$$ where m doesn't equal n, and both are odd. n is a fixed constant ...