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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4
votes
3answers
149 views

Closed form of $\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$

How would you recommend me to tackle the series $$\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$$? Can we possibly express it in terms of known constants? What do you think about it?
6
votes
2answers
140 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 ...
10
votes
2answers
227 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
votes
1answer
91 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
votes
3answers
107 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 $$\Phi(a,b)=\int_{0}^{1}\frac{dx}{(x^2+a^2)\sqrt{x^2+b^2}}$$
5
votes
1answer
222 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
vote
1answer
26 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
votes
1answer
134 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
vote
0answers
50 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
6
votes
3answers
138 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
vote
0answers
42 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
vote
3answers
110 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. ...
3
votes
1answer
38 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
vote
1answer
35 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
votes
2answers
75 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
38 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
vote
2answers
80 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
45 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
votes
1answer
101 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
votes
2answers
75 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
31 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
31 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
47 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
24 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
vote
1answer
34 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 ...
5
votes
0answers
70 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
37 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 ...
9
votes
2answers
107 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 $$\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$$
2
votes
1answer
57 views

Is it always true that no closed forms exists for any divergent 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
votes
0answers
22 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
35 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
33 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
146 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
105 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
172 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
99 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
249 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
126 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 ...
1
vote
1answer
56 views

Help in simplifying this nasty expression obtained after binomial expnasion

I have arrived to the following expression and was wondering if anyone can help me further simplify to something nicer, $$F= 1- [1-\text{exp} (- \alpha(N) ) ]^N= 1- \sum_{k=0}^{N} \binom{N}{k} ...
3
votes
2answers
157 views

Solving a 2 independent variables (2nd degree) recurrence relation

Changes to the recurrences and definition are changed! See here: $f(n, 1) = 2n^2 $ and $f (n, k) = 0$ for $k \geq 2n$ and for $k < 0$ and $f(n, 2n-1) = 1$ for all $n$. Question: Is it possible ...
7
votes
2answers
176 views

How to find closed-form of $\int_{0}^{+\infty} \operatorname{sech}^2 (x^2)\,dx$

How to find this integral closed form: $$I=\int_{0}^{+\infty}\operatorname{sech}^2{(x^2)}\,dx$$ where $\operatorname{sech}{(x)}$ is defined as secant of hyperbolic function. This problem ...
1
vote
0answers
49 views

New identity for sums of Bessel functions?

I've come across the following proposed identity: $$ ...
2
votes
3answers
47 views

Closed form for certain trigonometric integral

\begin{align}&\mbox{Is there a closed form for} \\[2mm]&\int_0^{\pi/2} \sin^{2}\left(\, nx\,\right)\sin\left(\, mx\,\right)\cot\left(\, x\,\right) \,{\rm d}x\ \quad\mbox{where}\quad m, n\ ...
0
votes
0answers
32 views

List of functions $\chi_{s,a}(n)$ defined on a Group such that $\chi_{s,a}(n)\in{s,a}$ and depending on the parity

Question Let $(G,\cdot,e)$ be a non-commutative group and $s,a \in G$ .I'm looking for interesting functions $\chi_{s,a}:\Bbb N \rightarrow G$ witht this property $$\chi_{s,a}(n)= \begin{cases} s, ...
3
votes
0answers
53 views

Closed form of a “harmonic” alternating dilogarithm sum

Does the following sum $$ S = \sum_{n\geq 2}(-1)^n \mathrm{Li}_2(2/n) = 1.14434\ 42096\ 91982\ 23727\ 39852\ 45805\ldots $$ have a closed form in terms of known constants? Neither the inverse ...
5
votes
2answers
78 views

An infinite exponential sum

I was trying to create a problem for a test I'm writing, and I ended up attempting to evaluate $$\sum_{n=1}^{\infty} \dfrac{1}{e^n-1}.$$ This definitely converges, but I have no idea how to go ...
1
vote
0answers
43 views

Zeros of $f$ in a disk

If $f$ holomorphic in a domain $U$ and $f(z)\neq 0$ for all $z\in U$ then every zero of $f$ is such that $f(q)=0$ and $\det(Df_{p})>0$. Using that I have to prove that if $f$ keeps that conditions ...
0
votes
1answer
63 views

How to solve integrals using series?

Many places I have seen when solving integrals you change a lot of it into sums. Finding $\int_{0}^{\pi/2} \dfrac{\tan x}{1+m^2\tan^2{x}} \mathrm{d}x$ Is just an example. So in general, how do you ...