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
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 ...
2
votes
1answer
45 views

Can you provide us a good approximation for $\sum_{n=1}^{\infty} \left| \log \left( 1+\frac{\mu(n)}{n^2} \right) \right|$?

Let $a_n=\frac{\mu(n)}{n^2}$, where $\mu(n)$ is the Möbius function. Since $\sum \left| a_n \right| $ is convergent by the comparison test, then a proposition from analysis ensures that $$\mathcal{S}=\...
3
votes
0answers
101 views

Closed form for $\int_0^1\frac{x}{\ln(x+1)(x^3+3x+3)}dx$

How can I evaluate the closed form of the following integral: $$\int_0^1\frac{x}{\ln(x+1)(x^3+3x+3)}dx$$ According to Wolfram Alpha, the numerical value of this integral is close to 0.2673, but ...
0
votes
2answers
53 views

Solution to recurrence relation, as a formula involving summation operator

Here is what I am tasked with.. Find a solution to the recurrence relation: $F(0) = 2$ $F(n+1) = F(n) + 2n^2 - 1$ as a formula involving the summation operator $$\sum_{i=1}^n$$ Sorry for the ...
2
votes
1answer
122 views

Can anyone verify $\int_{0}^{\infty}\frac{e^{-2nx}+2nx-1}{x(e^x+1)}dx=\ln{2n\choose n}$? [closed]

Central binomial coefficient from mathworld $$\frac{2^{2n+1}}{\pi}\int_{0}^{\infty}\frac{1}{(1+x^2)^{n+1}}dx={2n\choose n}$$ Here we have $\ln{2n\choose n}$ in term of another integral, $$\int_{0}...
4
votes
1answer
50 views

Analytic extension of $\sum_{k=1}^n\frac1k$ complex domain

The analytic extension: $$\sum_{k=1}^n\frac1k=\int_0^1\frac{x^n-1}{x-1}dx$$ I was wondering for what values of $n$ does this extend to, mainly complex values of $n$. I know it is defined for $n=0$, ...
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 ...
1
vote
3answers
34 views

Evaluate $\int_1^N \frac{-3N+6t-3}{t^3(N-t+1)^4}dt$ when $N=3$ or $N=5$

Let the Cauchy product $$(\zeta(3))^2=\sum_{n=1}^\infty c_n,$$ where $$c_n=\sum_{k=1}^n\frac{1}{k^3(n-k+1)^3},$$ and $\zeta(3)$ is the Apèry constant. Taking $f(x)=\frac{1}{x^3(N-x+1)^3}$ in Abel's ...
3
votes
2answers
198 views

A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$

Similar to the cfracs in this post, define the two complementary continued fractions, $$x=\cfrac{-(m+1)}{km\color{blue}+\cfrac{(-1)(2m+1)} {3km\color{blue}+\cfrac{(m-1)(3m+1)}{5km\color{blue} +\cfrac{...
4
votes
2answers
162 views

Evaluating the integral $\int \frac{x^2+x}{(e^x+x+1)^2}dx$

Evaluate $$\int \frac{x^2+x}{(e^x+x+1)^2}dx$$ I tried converting in the form of Quotient rule(seeing the square in the denominator), neither am I able to make the denominators' derivative in the ...
3
votes
0answers
94 views

Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...
1
vote
1answer
41 views

Integrate the Fourier Legendre by parts :$\int_{-1}^{1}\left( x^{2}-1\right) ^{m}\cos \pi x\:dx$

Having difficulty integrating the Fourier Legendre series by parts : $$\alpha_{m}=\int_{-1}^{1}\left( x^{2}-1\right) ^{m}\cos \pi x\:dx$$ I understand we can use the general formula : $$uv-\int ...
0
votes
0answers
23 views

Calculation of an integral involving the sum of a range of natural exponential functions

Does somebody know how to solve the following integral, I extremely hope I can obtain its close-form solution: \begin{equation} \int \sqrt{ \sum_{i=1}^{M}\sum_{j=1}^{M} e^{-\frac{\frac{\left|\mathbf{...
1
vote
1answer
22 views

Is there a general formula for the $n$'th variable of the solution for a lower triangular linear system of equations?

I have a countably infinite linear system of equations $Ax = b$, where $A$ is lower triangular with $-1$ at all diagonal entries, and $b = \{-1/2,0,0,...,0\}^T$. I.e the $n$'th unknown depends solely ...
0
votes
1answer
45 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 $$\left(\sum_{n=1}^\infty\frac{\log n}{e^n}\right)\left( \sum_{n=1}^\infty\frac{1}{e^n} \right)= \sum_{n=1}^\infty\sum_{k=1}^n\frac{\log k}{e^{k+n-k+1}},$$ ...
1
vote
1answer
36 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 ...
0
votes
1answer
119 views

Friend claims $\int_0^\infty\sum_{n=0}^{\infty}\frac{x^n}{2^{(n+1)^sx^{n+1}}+1}dx=\zeta(s+1)$?

My friend is making another claim on another integral! Can anybody verify it? Or his is mocking on me? Valid for all $s\ge1$ $$\int_0^\infty\sum_{n=0}^{\infty}\frac{x^n}{2^{(n+1)^sx^{n+1}}+1}dx=\...
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{...
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 ...
2
votes
3answers
92 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 ...
4
votes
1answer
148 views

How can we see that $ \sum_{n=0}^{\infty}\frac{2^n(1-n)^3}{(n+1)(2n+1){2n \choose n}}=(\pi-1)(\pi-3) $?

I wonder will it help me so prove it if I was to decompose it into partial fractions? Mathematica approves of the identity; it is converges. can anyone help me to prove it? $$ \sum_{n=0}^{\infty}\...
2
votes
2answers
75 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? ...
0
votes
1answer
38 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 $n$: $$ \sum_{t=-\infty}^{\infty}e^{-\beta(t+\alpha)^{2}}H_{n}(t+\alpha) $$ I'...
0
votes
1answer
115 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 + \sum_{i=0}^n{...
5
votes
1answer
190 views

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

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 \equiv \...
2
votes
1answer
27 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
46 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}} e^{-...
-2
votes
2answers
54 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
vote
1answer
44 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 ...
3
votes
1answer
40 views

Solving the recursion $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$

Solving the recursion $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$ $p_n = p \cdot p_{n-2} + (1-p)p_{n-1}$ $p_n = p \cdot p_{n-2} +p_{n-1} - p\cdot p_{n-1}$ $p_n - p_{n-1} = (-p)(p_{n-1} - p_{n-2})$ $= (-...
5
votes
2answers
147 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 of ...
7
votes
2answers
82 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 this....
1
vote
1answer
38 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} \\ &= \sum\limits_{n=-\...
3
votes
2answers
91 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
117 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 ...
0
votes
1answer
81 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
77 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 ...
14
votes
1answer
119 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 ...
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
169 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 \frac{\ell(w_{i-1})}{...
2
votes
1answer
33 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.
1
vote
0answers
89 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
60 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. ...
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 ...
6
votes
2answers
114 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
votes
0answers
66 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 ...
1
vote
1answer
40 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) ...
1
vote
0answers
71 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, ...
33
votes
1answer
756 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
84 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 ...