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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Packing an infinite sequence of disks

Let $r > 1$ and $Q(r)$ denote the supremum of values of $q$ such that an infinite sequence of disks, whose radii form an infinitely decreasing geometric progression with the start value $1$ and the ...
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3answers
137 views

What is the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}$?

A while ago, I started to look at expressions of the following form: $$ S_p:=\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}, $$ where $p$ is prime, because otherwise things get too complicated for me at ...
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3answers
44 views

Closed-form term for $\sum_{i=1}^{\infty} i q^i$ [duplicate]

I am interested in the following sum $$\sum_{i=1}^{\infty} i q^i$$ for some $q<1$. Is there a closed-form-term for this? If yes, how does one derive this? I am also interested in ...
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0answers
26 views

Closed-form solution to polynomials of this special form?

I have two sets of real numbers $\{a_k\}$ and $\{b_k\}$ such that: $$a_n \leq \ldots \leq a_1 < b_1 \leq \ldots \leq b_n$$ Now consider the following polynomial equation: $$\prod_{k=1}^n (x - ...
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2answers
47 views

closed form for $\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$

$$\int_0^1x^{a+1}(1-x^2)^bJ_a(cx)dx$$ my friend posted the integral on the fb and I tried to solve it but I faild because I have little information about bessel function so could some one help ?
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4answers
72 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
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1answer
65 views

Closed form of $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$

Recently, I came across the following exercise on the course of discrete math Find a closed form for $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$ So I tried some of the usual techniques: Let ...
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0answers
52 views

Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$ where $\lambda>0$, $a>0$, $d>0$ and where $b$, ...
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66 views

Is there a closed form solution to this equation?

I have the following equation $e^{-x/b}(a+x) = e^{x/b}(a-x)$ where $b > 0$, and $a > 0$ I need to solve for $x$. I can do it numerically, but would prefer if there was a closed form solution. ...
2
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2answers
97 views

Closed form for $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$?

Let $x>1$ and $f(x)=\int_{0}^{+\infty}e^{it^{x}}dt$. Does this integral have a closed form ? Fist point, the integral converges. Indeed let $u=e^{it^{x}}$ and $v=\frac{-i}{x}t^{1-x}$ we have ...
5
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3answers
163 views

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$

I'm look for a closed form evaluation of the following improper definite integral involving logarithms: $$\begin{align} I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\ ...
2
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1answer
29 views

Does $\sum_{i=1}^{k-1}\lceil \log_2\frac{N}{i}\rceil$ have a closed form?

Does the following have a closed formula? $$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
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1answer
101 views

Proving that $\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

How can we prove, without employing the aid of residues or various transforms, that, for $n>2$ $$\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$$ Motivation: ...
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1answer
189 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
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1answer
208 views

Is there a closed form expression for the sum of all the proper divisors of an integer?

I have already found a summation formula here: http://math.stackexchange.com/a/22723, and also a very interesting recursive formula here: http://math.stackexchange.com/a/22744. Any ideas on how to ...
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0answers
82 views

What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form?

In a consideration of summation of a series $$ s = a_0 + a_1 + a_2 + \cdots \tag 1$$ with $$\lim_{k \to \infty} a_k=0$$ but slowly decreasing, the coefficients $a_k$ are somehow related to $1/k^2$ ...
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30 views

Help getting a closed-form solution to a maximisation problem

I'm working through a maximisation problem that I can't seem to get a closed-form solution to. It may be the case that there is no closed-form solution, but I would like a second opinion, since I've ...
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4answers
565 views

Evaluate $\int_{0}^1 \prod_{k=2}^n \lfloor kx \rfloor dx$

Let $n\ge2$ be an integer , then $$\int_{0}^1 \prod_{k=2}^n \lfloor kx \rfloor dx=\text{ ?}, $$ where $\lfloor \space \rfloor$ is the "floor-function"
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5answers
315 views

Find a closed expression for a formula including summation

Let: $$\sum\limits_{k = 0}^n {k\left( {\matrix{ n \cr k \cr } } \right)} \cdot {4^{k - 1}} \cdot {3^{n - k}}$$ Find a closed formula (without summation). I think I should define this as a ...
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2answers
271 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\, ...
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Seeking closed-form solution to $\sum_{n=1}^{\infty}\frac{\log{(1+n)}}{(1+n)^{\alpha}-1}$

I'm looking for a closed-form solution to this infinite series: $$S(\alpha):=\sum_{n=1}^{\infty}\frac{\log{(1+n)}}{(1+n)^{\alpha}-1},~~~\Re(\alpha)>1.$$ My attempt All I've really been ...
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2answers
44 views

Summation closed form.

I have tried to figure out how to get the close form of: $$\sum_{k=0}^n k^22^{n-k}.$$ I tried to write down each number of the summation but couldn't find any thing to do with that.. can please ...
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1answer
10 views

How to determine a deficit of base 10 exponent from a minimum unit

I'm trying to determine if a number includes a portion in deficit of a base 10 exponent from a minimum unit. For example, given a minimum unit of $0.01$, $1$ would pass the test, but $1.001$ would ...
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0answers
62 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
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0answers
11 views

Compact closed form for linear recurrence formulas

Assume you have some linear recursion formula $$f(\vec x)=\sum_{\vec y\in Y}w_{\vec y}f(\vec x - \vec y)$$ Where $\vec y\geq 0 $ and $||\vec y||>0$, $w_{\vec y}\in\mathbb{R}$ and $\vec x , \vec ...
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3answers
191 views

How to evaluate improper integral $\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}dx$?

I'm trying to evaluate the improper integral, $$I(a)=\int_{0}^{\infty}\frac{\tan^{-1}{x}}{e^{ax}-1}\mathrm{d}x,~~~\text{where }a\in\mathbb{R}^+.$$ Does this integral have a simple closed form ...
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1answer
69 views

Closed form of a trigonometric integral sought

I am trying to evaluate the definite integral $I(a,b)$, with $a,b\in\mathbb{R}$, defined by $$I(a,b):=\int_{0}^{2\pi}\sqrt{1-(a+b\cos{\theta})^2}\mathrm{d}\theta.$$ Assume $a,b$ are suitably ...
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2answers
112 views

Superelliptic Area Of $x^5+y^5=r^5$

$${\LARGE\int}_0^\tfrac\pi2\frac{dx}{\bigg(\sqrt[{\Large 5}]{\cos^5x+10\cos^3x\sin^2x+5\cos x\sin^4x}\bigg)^{\large 2}}~=~?$$ Its numerical value is about $1.40171345128228$. Maple, Mathematica, ...
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1answer
36 views

Does sequences related to function for $lcm(1,2,3 \cdots n)$ exists?

This just came out of curiosity let $$L(n)=lcm(1,2,3 \cdots n)$$ and I know that we can write this with the help of some product involving primes and all . But what I am interested is in Does ...
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1answer
112 views

Compute $\sum_{n=1}^{+\infty}\frac{\mathrm{e}^{-\sqrt{n}}}{\sqrt{n}}$

Does the following series have a closed form ? $$\sum_{n=1}^{+\infty}\frac{\mathrm{e}^{-\sqrt{n}}}{\sqrt{n}}$$ Motivation : The original exercise is Compute $\int_{1}^{+\infty} ...
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2answers
356 views

Evaluating $\int_0^{2\pi}\frac{dt}{\sqrt[4]{P(\cos t,\sin t)}}$

$${\LARGE\int}_0^{2\pi}\frac{dt}{\sqrt[{\LARGE 4}]{A\Big(\sin^8t+\cos^8t\Big)+B\Big(\sin^6t\cos^2t+\sin^2t\cos^6t\Big)+C~\sin^4t\cos^4t}}~=~?$$ where $A=0.3$, $B=-3.3$, and $C=10$. Its numerical ...
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84 views

A logarithm integral

Calculate the integral \begin{align} \int_{0}^{1} \frac{ \ln(\sqrt{x} - \sqrt{1-x}) }{ \sqrt{x} } \ dx \end{align} and show the value is negative.
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2answers
52 views

$x^x-x+5=\frac{29}{4}$

A friend of mine is claiming to have a closed form solution to $x^x-x+5=\frac{29}{4}$, plotting it into wolfram alpha gives an approximation, and the equation doesn't seem very easy to solve. Can any ...
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2answers
165 views

Harmonic Numbers series I

Can it be shown that \begin{align} \sum_{n=1}^{\infty} \binom{2n}{n} \ \frac{H_{n+1}}{n+1} \ \left(\frac{3}{16}\right)^{n} = \frac{5}{3} + \frac{8}{3} \ \ln 2 - \frac{8}{3} \ \ln 3 \end{align} where ...
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0answers
44 views

Dilogarithm in closed form

Is there a closed form expression for \begin{align} e^{\Large\frac{i\pi}3} \text{Li}_{2}\left( \frac{e^{\Large\frac{i\pi}3} }{2}\right) + e^{-\Large\frac{i\pi}3} \text{Li}_{2}\left( ...
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1answer
72 views

Summing up numbers from the continued fraction of $e ^ \pi$ and $\pi ^e$

I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting ...
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36 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
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1answer
63 views

May I know that there is a special structure or solution on this linear fractional optimization?

I am not familar with the optimization problems, but I want to know a very simple formulation: \begin{array}{cc} {{\max_\mathbf{x}}} & \frac{\mathbf{a}^{T}\mathbf{x}}{\mathbf{b}^{T}\mathbf{x}}\\ ...
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3answers
127 views

Do these integrals have a closed form? $I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$

The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign? $$I_1 = \int_{-\infty ...
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0answers
69 views

Log Log Integrals III

The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( ...
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49 views

Closed form definite integral. [closed]

I have a definite integral that I am trying to solve. Any hint or reference is urgently sought $$\int_0^\infty \nu y^r e^{\nu y}\{1+\psi(e^{\nu y}-1)\}^{-\tfrac1\psi -1}dy.$$ where $r$ is any ...
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1answer
52 views

What is this waveform?

Consider the following infinite series: $\text{f} \left( x \right) =\displaystyle \sum \limits_{n=1}^{\infty} \frac {\sin \left( n x\right)}{n^2}$ We know that $\text{f} \left( x \right)$ is ...
9
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3answers
177 views

Definite integral - closed form: $\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x$

I'm struggling with this definite integral: $$ \int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x. $$ Any help will be greatly appreciated.
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2answers
95 views

Log Log Integrals

Evaluate the integrals \begin{align} I_{1} &= \int_{0}^{1} \ln^{2n}(x) \ \ln\left(\ln\left(\frac{1}{x}\right)\right) dx \end{align} and \begin{align} I_{2} = \int_{0}^{1} \ln^{2n}(x) \ ...
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1answer
62 views

Finding the area enclosed by Batman curve

I was reading this thread: Is this Batman equation for real? and I wondered if the area bounded by the curve had a nice closed form. The resulting integrals were slightly beyond me, so I thought I ...
0
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2answers
65 views

Closed form of a sequence containing the pattern $\{+,-,-,+,+,-,-,…\}$

I was in the middle of answering this question that asked how to get the Taylor series of $cos(x)$ centered at $a=\frac{\pi}{3}$. I was reaching the point where I was about to write $(-1)^n$ making ...
1
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1answer
46 views

Matrix exponential of a simple bidiagonal matrix

I am interested in finding an expression (closed form or recursive) for the matrix exponential of this banded matrix: $$ \begin{pmatrix} 0 & 1 & 0 & 0 & \cdots & 0 & 0 ...
0
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0answers
29 views

Simple representation of a Sum

I have a probably pretty simple question. if I have $\sum_{i=0}^{n} 4^i$ and I want a closed representation, Wolfram Alpha gives me: $4^{n+1}/3-1/3$ Why is that? How do I get to that? Thanks!
1
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1answer
59 views

Is there closed form for $(1-p)(1-p^2)(1-p^3)…$ or its Taylor expansion?

I was considering the following problem: Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and ...
0
votes
1answer
39 views

Quadratic differential equation - closed form solution?

Can a solution for x(t) be found from the following real valued differential equation $$a\frac{d}{dt}\!(x(t))^2 + x(t) +b\frac{d}{dt}\!(y(t)) = 0$$ in terms of only y(t), it's integrals or ...