All Questions
176
questions
2
votes
2
answers
138
views
A question about sum with reciprocal quartic
Evaluate
$$
\sum_{n=1}^{\infty} \frac{n+8}{n^{4}+4}
$$
According to WolframAlpha it is $\pi\coth(\pi) - \dfrac{5}{8}$
My attempt:
I tried to separate $\dfrac{n}{n^{4}+4}$ and $\dfrac{8}{n^{4}+4}$. ...
- 192
4
votes
2
answers
200
views
How to evaluate this sum $\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$
How to evaluate this sum $$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
My attempt
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{(n^2 + 3n + 1)(n^2 - 3n + 1)}$$
$$= \sum_{n=1}^{\infty} \...
1
vote
0
answers
188
views
Closed form for $A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ ...
- 16.4k
2
votes
0
answers
141
views
closed form for limit?
Consider the function
$$ f(x)=\lim_{k \to \infty}\bigg(\int_0^x \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)\bigg( \int_0^1 \sum_{n=1}^k e^{\frac{\log n}{\log r}}~dr \bigg)^{-1} $$
I want to find ...
- 756
2
votes
1
answer
151
views
For $J=\{1,2,\dots,n \}$ is there an easy way to compute $\prod\limits_{i\in J | i \ne k} (k-i)$?
When I studied calculus at my university there is one question that I hated the most which is given a finite number of terms for some sequence find the $n-$th term. I hated this type of question ...
- 6,620
2
votes
0
answers
70
views
Closed form for ${_3F_2}\!\left(\begin{array}c\tfrac34,1,1\\\tfrac32,\tfrac74\end{array}\middle|1\right)$
I am trying to find the closed form the expression $${_3F_2}\!\left(\begin{array}c\tfrac34,1,1\\\tfrac32,\tfrac74\end{array}\middle|1\right).$$ I was able to convert the expression into the series $${...
-1
votes
2
answers
96
views
Evaluate $\sum_{m=1}^{\infty} \frac{(-1)^m \mathcal{H}_{2m} }{2m+1} - \frac{1}{2} \sum_ {m=1}^{\infty} \frac{(-1)^m \mathcal{H}_m}{2m+1}$ [duplicate]
Let's declare $\mathcal{G}$ is constant of Catalanand the $\mathcal{H}_m-st$ mharmonic term. Let it be shown that:
$$\displaystyle{\sum_{m=1}^{\infty} \frac{(-1)^m \mathcal{H}_{2m} }{2m+1} -\frac{1}{2}...
1
vote
2
answers
105
views
Evaluate $\sum_{n=1}^{\infty} (-1)^{n+1} H_n \left( \frac{1}{n+1} - \frac{1}{n+3} + \frac{1}{n+5} - \ldots \right)$
$$\sum_{n=1}^{\infty} (-1)^{n+1} H_n \left( \frac{1}{n+1} - \frac{1}{n+3} + \frac{1}{n+5} - \ldots \right) = \frac{\pi}{16} \cdot \log(2) + \frac{3}{16} \cdot \log(2) - \frac{\pi^2}{192}$$
$$\sum_{k=...
3
votes
1
answer
143
views
Show that $\sum_{n=1}^{\infty} \frac{(-1)^n (\psi(n) - \psi(2n))}{n} = \frac{\pi^2}{16} + \left(\frac{\ln(2)}{2}\right)^2$
Show That
$$\sum_{n=1}^{\infty} \frac{(-1)^n (\psi(n) - \psi(2n))}{n} =\bbox[15px, #B3E0FF, border: 5px groove #0066CC]{\frac{\pi^2}{16} + \left(\frac{\ln(2)}{2}\right)^2}$$
my work
$$\sum_{n=1}^{\...
10
votes
3
answers
190
views
Show that $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^2} \sum_{n=1}^k \frac{1}{n}=\frac{5\zeta(3)}{8}$
$$\sum_{k=1}^{\infty} \dfrac{(-1)^{k+1}}{k^2} \sum_{n=1}^k \dfrac{1}{n}=\frac{5\zeta(3)}{8}$$
I tried to create a proof from some lemmas some are suggested by my Senior friends
Lemma 1 $$
{H_n} = \sum\...
user1254447
2
votes
0
answers
82
views
Limit : $\lim_{n\to+\infty}a^n(n-\zeta(2)-\zeta(3)-\cdots-\zeta(n))$
question
Compute the limit $$\displaystyle{\lim_{n\to+\infty}a^n(n-\zeta(2)-\zeta(3)-\cdots-\zeta(n))}$$, if any, for the various values of the positive real a, where $\zeta$ the zeta function of Mr. ...
6
votes
2
answers
185
views
Calculate $\sum\limits_{n = - \infty }^\infty {\frac{{\log \left( {{{\left( {n + \frac{1}{3}} \right)}^2}} \right)}}{{n + \frac{1}{3}}}} $
question:
how do we find that:
$$ S = \sum\limits_{n = - \infty }^\infty {\frac{{\log \left( {{{\left( {n + \frac{1}{3}} \right)}^2}} \right)}}{{n + \frac{1}{3}}}} $$
I modified the sum
$$\sum\...
user1254447
0
votes
1
answer
74
views
Closed form solution for partial summation of $\sum_{x=1}^{k} \frac{2^{\frac{1}{x}}}{{x^2}}$
Recently I've been working on solving summations and I found this one to be quite tricky.
$\sum_{x=1}^{k} \frac{2^{\frac{1}{x}}}{{x^2}}$
The integral which this is based off of, can be solved with u ...
5
votes
2
answers
211
views
Prove $_4 F_3\left(2,\frac32,\frac32,\frac32;\frac52,\frac52,\frac52;1\right)=\frac{27}{16}\left(\pi^2-7\zeta(3)\right) $
How to prove the following result about the generalized hypergeometric function $_4 F_3$?
$$_4 F_3\left(2,\frac32,\frac32,\frac32;\frac52,\frac52,\frac52;1\right)\stackrel{?}=\frac{27}{16}\left(\pi^2-...
- 3,392
0
votes
1
answer
88
views
A closed formula for $\prod_{n=1}^{N}(an-b)\;$? [closed]
Is there a way to express a closed formula for the finite product:
$$\prod_{n=1}^{N}(an-b)$$
Maybe it can be done through the Pochhammer symbol?
- 53