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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1answer
29 views

How to find a recursive formula for some sequence

I know how to find a non-recursive formula for a recursively defined sequence. However, now I have this puzzle which gives me a sequence (but not the recursive definition) and challenges me to find ...
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5answers
224 views

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

Today I discussed the following integral in the chat room $$\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$$ where $0\leq a, b\leq \pi$ and ...
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3answers
33 views

Finding a closed-form formula for a sequence that is defined recursively

$$a_0 = 0, a_1 = 1 \quad \text{ and } \quad a_n = a_{n-1} + 2a_{n-2}\quad \text{ for }n\geq 2$$ a) Find $a_2,a_3,a_4,a_5$ b) Find a closed form-formula for $a_n$ I found the value to be ...
0
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2answers
61 views

Closed form of a series

Is there exist a closed form for the series of the form $$ \sum_{k=0}^{[n/2]}(-a)^{k}\binom{n-k}{k} $$ where $0<a\leq1$. For example, we have $$ ...
5
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3answers
160 views

How to evaluate $\int_{0}^{\infty}\frac{(x^2-1)\ln{x}}{1+x^4}dx$?

How to evaluate the following integral $$I=\int_{0}^{\infty}\dfrac{(x^2-1)\ln{x}}{1+x^4}dx=\dfrac{\pi^2}{4\sqrt{2}}$$ without using residue or complex analysis methods?
1
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0answers
42 views

Proving $\displaystyle{\int \frac{x^3}{{\rm e}^x - 1}}dx$ cannot be evaluated in closed form in terms of elementary functions

It is known the following indefinite integral $$\int \frac{x^3}{{\rm e}^x - 1} dx$$ cannot be evaluated in closed form in terms of any of the elementary functions of mathematics. A proof of this can ...
1
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2answers
75 views

Is it possible to evaluate $\int_0^1 \sin(\frac{1}{t})\,dt\,$?

I was wandering if it possible to evaluate the value of the following improper integral: $$ \int_0^1 \sin\left(\frac{1}{t}\right)\,dt $$ It is convergent since $\displaystyle\int_0^1 ...
4
votes
1answer
167 views

Evaluating $\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$

The title says it all, really - I am looking for $$\int \arccos\left(\frac{\cos(x)}{r}\right) \, \mathrm{d}x$$ where $0<r<1$ and $x$ is in a domain where the integrand is real. It came up ...
4
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0answers
58 views
+100

Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$

Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} ...
1
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0answers
31 views

Closed forms for two times series similar to geometric series, but with additional power

Does anyone know a close form solutions to any of the following time series? (approximate upper bounds might as well work). $$ \sum_{k=1}^T \frac{1}{2^{k^2}} $$ or $$ \sum_{k=1}^T k ...
13
votes
2answers
311 views

A couple of definite integrals related to Stieltjes constants

In a (great) paper "A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations" by Iaroslav V. Blagouchine, the following ...
6
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2answers
122 views

Show $\sum_{n=1}^\infty\frac{1}{n^2+3n+1}=\frac{\pi\sqrt{5}}{5}\tan\frac{\pi\sqrt{5}}{2}$.

How to show that $$\sum_{n=1}^\infty\frac{1}{n^2+3n+1}=\frac{\pi\sqrt{5}}{5}\tan\frac{\pi\sqrt{5}}{2}$$ ? My try: We have ...
7
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2answers
164 views

Closed-form of $\sum_{n=0}^\infty\;(-1)^n \frac{\left(2-\sqrt{3}\right)^{2n+1}}{(2n+1)^2\quad}$

The following question is purely my curiosity. During my calculation to answer @Chris'ssis's question in chat room I encountered this series $$\sum_{n=0}^\infty\; ...
2
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1answer
44 views

Sum of exponential functions involving powers of two

I came across a weird series with exponential functions and powers of two: $$\sum_{k=0}^{\infty} \left(1 - e^{-2^{-k}z} \right), z \in \mathbb R_+$$ and have no idea how to solve this (if there even ...
5
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3answers
84 views

Closed form of $\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}$

Is it possible to get a closed form of the following integral $$\Phi(a,b)=\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}$$
4
votes
1answer
165 views

Closed-form of $\int_0^{\pi/2}\frac{\sin^2x\arctan\left(\cos^2x\right)}{\sin^4x+\cos^4x}\,dx$

I have just seen two active posts about integrals of inverse trigonometric function, $\arctan(x)$, here on MSE. So I decide to post this question. This integral comes from a friend of mine (it's not a ...
1
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1answer
25 views

Find a closed form

How do I prove (with strong induction) that every positive integer $n$ has a representation in the form $$n = c_r2^r + c_{r−1}2^{r−1} + \cdots + c_2 2^2 + c_1 2 + c_0$$ where $r$ is a nonnegative ...
3
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1answer
126 views

Compute$\int\limits_{0}^{2} \sqrt{x^2-2x+2}\ln(2+x)dx$.

Compute: $I=\displaystyle \int\limits_{0}^{2} \sqrt{x^2-2x+2}\ln(2+x)dx$. I tried to : $I=\displaystyle \int \limits_{-1}^{1}\sqrt{t^2+1}\ln(3+t)dt$ set $t=\tan u\Rightarrow dt=(1+\tan^2u)du$ and ...
1
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0answers
47 views

Closed form of an equation

How could I find a closed form for the equations 1^3 = 1 , 2^3 = 3 + 5 , 3^3 = 7 + 9 + 11 , 4^3 = 13 + 15 + 17 + 19, 5^3 = 21 + 23 + 25 + 27 + 29 ... and Prove this closed form by induction? Thanks
4
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2answers
82 views

Show that $\sum\limits_{i=0}^{n/2} {n-i\choose i}2^i = \frac13(2^{n+1}+(-1)^n)$

While doing a combinatorial problem, with $n$ being even, I came up with the expression $$\sum_{i=0}^{n/2} {n-i\choose i}2^i$$ for which I used wolfram to get a closed form expression of ...
1
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0answers
35 views

Closed-form expressions of $\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$

Does anybody know if there's a closed-form expression of this series? $$\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$$ where $a$ and $b$ are strictly positive. It's easy to see that it's ...
1
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3answers
106 views

Find a closed form for the equations $1^3 = 1$, $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$

This is the assignment I have: Find a closed form for the equations $1^3 = 1$ $2^3 = 3+5$ $3^3 = 7+9+11$ $4^3 = 13+15+17+19$ $5^3 = 21+23+25+27+29$ $...$ Hints. ...
2
votes
1answer
22 views

Closed Form Solution for Recurrence Relation

Is it possible to calculate the closed form solution for the following recurrence relation? $$ T(n) = T\left(\frac{n}{2}\right) + T\left(\frac{n}{2} + 1\right) + \frac{n}{2} $$ I am trying to teach ...
1
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1answer
34 views

I suspect this integral has a closed form but I can't find it

$$\int_{-\infty}^\infty \!\!\text{d} r\dfrac{1}{r}e^{\frac{-(r-r_0)^2}{\delta^2}}\sin(k r)$$ Where $\delta>0$, $r_0\in \mathbb{R}$. Can anyone help me with this? it seems to me there has to be a ...
0
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2answers
73 views

Fixing the closed form of $\sum_{k=1}^nk\sin^2(kx).$

I've been working on finding the closed form of this:$$\sum_{k=1}^nk\sin^2(kx).$$ Using the fact that:$$\sum_{k=1}^nku^k={u\over (1-u)^2}\bigg[nu^{n+1}-(n+1)u^n+1\bigg]\forall u\ge 1\quad (1)$$ I ...
2
votes
0answers
24 views

Simplified expression of $ _2F_1((K-1)a,K,Ka,x) $

Is there any simplified expression of this Hypergeometric function $ _2F_1((K-1)a,K,Ka,x) $ Thanks!
1
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2answers
63 views

Trying to find the closed form for the nth term of $\frac{1}{1-x^4}$

I know that $\frac{1}{1-x^4}$ is the generating function for the sequence (1, 0, 0, 0, 1, 0, 0, 0, 1, ...) I don't know how to find the closed form for the nth term though. Itried messing around with ...
-2
votes
1answer
41 views

Evaluating the series with arctangents: $\sum_{r=1}^\infty \tan^{-1}\frac{2r}{2+r^2+r^4}$

If $$S=\sum\limits_{r=1}^\infty\tan^{-1}\left(\frac{2r}{2+r^2+r^4}\right)$$ then what is cot S? Options: A) 1; B) 3; C) 1/3; D) 2 Does it converge? I don't really know how to find the ...
4
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2answers
89 views

How solve $\int \frac{dx}{(x^2-x)^x}$ [closed]

I want solve $$\int \frac{dx}{(x^2-x)^x}$$. thanks for help
0
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2answers
69 views

How to find the generating function and the closed form for the generating form

I'm trying to find the generating function and the closed form for the generating form for this sequence: $0,1,-2,4,-8,16,-32,64...$ I've tried the following: I think it's an index shift so that's ...
1
vote
1answer
25 views

Is a finite continued fraction a closed-form expression?

We had a discussion regarding this answer on Electrical Engineering. The answer in question discussed a finite continued fraction. We're wondering whether it's a closed-form expression or not. ...
3
votes
1answer
29 views

Closed form for $x+2^ax^2+3^ax^3+…+n^ax^n$

I was wondering if there was a closed form for $$f(x)=x+2^ax^2+3^ax^3+...+n^ax^n+...$$ I have tried to find one but I had no luck. If you divide by $x$ and then integrate you get ...
3
votes
2answers
42 views

How does the recursion relation work in the solution to this differential equation (using series)?

Sorry for the vague title but it would not let me post the first step and last step of this equation (too many characters!). How does $$\dfrac{a_0}{3n(3n-1)(3n-3)(3n-4)\cdots 9 \cdot 8 \cdot 6 \cdot ...
1
vote
1answer
16 views

Find a closed form equation of the following sequence: ${0,0,-2,0,4,0,-6,…}$

Find a closed form equation of the following sequence: ${{0,0,-2,0,4,0,-6,...}}$ I know $1+-1^n$ = 0 if n is odd and 1 if n is even. However finding alternating signs when plugging in only even ...
1
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1answer
28 views

Is there a closed form to $a_{n+2}=\frac{(n+1)(n-2)a_{n+1} + (4n+3)a_n - a_{n-1}}{(n+2)(n+1)}$ in terms of $a_0$ and $a_1$?

Is there a closed form solution to $$a_{n+2}=\dfrac{(n+1)(n-2)a_{n+1} + (4n+3)a_n - a_{n-1}}{(n+2)(n+1)}$$ that can be written in terms of $a_0$ and $a_1$ given the fact that that $$a_2 = \dfrac{2a_1 ...
3
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0answers
51 views

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

Is there a possibility to find a closed-form for $$\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$$ We have $$I=\int_0^1\frac{Li_2^3(-x)+x^4Li_2^3(-\frac{1}{x})}{x^3}\,dx$$ After repeatedly ...
2
votes
1answer
33 views

Closed-form of prime zeta values

The prime zeta function is defined as $$P(s)=\sum_{p\,\in\mathrm{\mathcal P}} \frac{1}{p^s},$$ where $\mathcal P$ is the set of prime numbers. It converges for all $\Re(s)>1$. There is a related ...
8
votes
2answers
91 views

Closed- form of $\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$

Is there a possibility to find a closed-form for $$\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$$
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0answers
20 views

General question: Algorithm or procedures to show existence of closed forms of any infinite series?

Having seen many questions regarding finding closed form of integrals or infinite series, and some users providing either the final answer or detailed solution, and also reading how one finds a closed ...
2
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0answers
17 views

Closed-form expectation of CES function of a random variable?

I am faced with the following function, called CES (constant elasticity of substitution), of the continuously-distributed random variable $\epsilon$: $f(\epsilon) = (a^\sigma + ...
4
votes
1answer
32 views

$(1-t^2)\frac{\mathrm{d}^2y}{\mathrm{d}t^2}-t\frac{\mathrm{d}y}{\mathrm{d}t}+(a+2q (1- 2t^2))y=0$

So I have to solve $$(1-t^2)\frac{\mathrm{d}^2y}{\mathrm{d}t^2} -t\frac{\mathrm{d}y}{\mathrm{d}t}+(a+2q (1-2t^2))y=0$$ All substitutions seem to fail, some trigonometric ones fail less than the rest, ...
5
votes
1answer
31 views

Conditional iterations constant.

Let $f(0)=2.$ Define for positive integers $n$ : $f(n+1) = \frac{3}{2} f(n)$ if $f(n)$ is even. $f(n+1) = \frac{3}{2}(f(n)+1)$ if $f(n)$ is odd. We now have $\lim_{n->\infty} \dfrac{4* (3/2)^{n} ...
9
votes
1answer
105 views

Sum of Harmonic numbers $\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$

Finding the closed form of: $$\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$$ where, $\displaystyle H_n^{(2)} = \sum\limits_{k=1}^{n}\frac{1}{k^2}$ It appears when we try to determine the ...
1
vote
2answers
45 views

How find the following integral?

I want find a closed Form for below integral$$\int \frac{1}{-1-aX+\frac{1}{2}bX^2} dX$$. thanks for help
0
votes
2answers
32 views

Find a closed form of $\sum_{i=0}^{n}\frac{x^i}{\left(1-x^2\right)^i}$.

Let $\displaystyle f(x) = \sum_{i=0}^{n}\dfrac{x^i}{\left(1-x^2\right)^i}$ While solving a problem I came up with this function which requires me to solve this function into a closed form. How do I ...
5
votes
2answers
101 views

How to solve $\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)} dx\,?$

I encountered this integral and tried to solve it. As you can expect I could not solve this and thought I will ask it here. The integral is: $$\int_{0}^{2\pi} \frac{\cos(50x)}{5+4\cos(x)}\, dx$$ I ...
11
votes
1answer
145 views

An integral $\int^\infty_0\frac{\tanh{x}}{x(1-2\cosh{2x})^2}{\rm d}x$

I would like to enquire about the possible methods of computing the following integral $$ \color{blue}{% \int^{\infty}_{0}\frac{\tanh\left(\, x\,\right)} {x\left[\, 1 - 2\cosh\left(\, ...
6
votes
3answers
96 views

what's the summation of this finite sequence?

$a$ and $b$ are positive integers. The summation is $$\sum\limits_{x = 1}^a {x\left( {\begin{array}{*{20}{c}} {a + b - x}\\ b \end{array}} \right)} .$$ Any closed-form expression? I thought it ...
5
votes
1answer
212 views

Closed-form formula for the $n^{\rm th}$ term of ${1,1,1,1,\ldots, 1}, {2,2,2,2,\ldots, 2},\ldots, {k-1, k-1}, k.$

Let $k$ be a positive integer. Consider a finite sequence $L_k(n)$ given by $$\underbrace{1,1,1,1,\ldots, 1}_{k\text{ terms}}, \underbrace{2,2,2,2,\ldots, 2}_{k-1\text{ terms}},\ldots, ...
5
votes
1answer
125 views

General Solution of $y'(x)+p(x)e^{r(x) y(x)}=q(x)$

I solved the case for the non-homogenous constant coefficients case and I wondered if there is a way to find a general solution to a non-constant coefficient case. I don't know how to approach this at ...