Linked Questions

20
votes
6answers
9k views

Showing $ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}}$

I would like to show that: $$ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\frac{\pi}{3\sqrt{3}} $$ We have: $$ \sum_{n=0}^{\infty} \frac{1}{(3n+1)(3n+2)}=\sum_{n=0}^{\infty} \frac{1}{3n+1}-\frac{1}{...
36
votes
5answers
7k views

Show $\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$

How to show the following equality? $$\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$$
26
votes
2answers
1k views

Proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$

Show that $$\sum_{n=1}^{\infty}\frac{\sinh\big(\pi n\sqrt2\big)-\sin\big(\pi n\sqrt2\big)}{n^3\Big({\cosh\big(\pi n\sqrt2}\big)-\cos\big(\pi n\sqrt2\big)\Big)}=\frac{\pi^3}{18\sqrt2}$$ I have no ...
11
votes
4answers
989 views

Closed form for $\sum_{n=-\infty}^{\infty} \frac{1}{n^4+a^4}$

I've been working with series this week and came across a couple that have been bugging me. I'm looking for the closed form of: $$ J(a)=\sum_{-\infty}^{\infty} \frac {1} {n^4+a^4} $$ As with the ...
6
votes
7answers
502 views

Prove that $\int_{-\infty}^{+\infty} \frac{1}{x^4+x^2+1}dx = \frac{\pi}{\sqrt{3}}$

Good evening everyone, how can I prove that $$\int_{-\infty}^{+\infty} \frac{1}{x^4+x^2+1}dx = \frac{\pi}{\sqrt{3}}\;?$$ Well, I know that $\displaystyle\frac{1}{x^4+x^2+1} $ is an even function ...
5
votes
3answers
2k views

Proving of $\frac{\pi }{24}(\sqrt{6}-\sqrt{2})=\sum_{n=1}^{\infty }\frac{14}{576n^2-576n+95}-\frac{1}{144n^2-144n+35}$

This is a homework for my son, he needs the proving.I tried to solve it by residue theory but I couldn't. $$\frac{\pi }{24}(\sqrt{6}-\sqrt{2})=\sum_{n=1}^{\infty }\frac{14}{576n^2-576n+95}-\frac{1}{...
4
votes
3answers
315 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?
7
votes
2answers
236 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 $$n+3n+1=\left(n+\frac{3+\sqrt{5}}{2}\right)\left(n+\frac{3-\sqrt{5}}{2}\...
4
votes
2answers
745 views

Using contour integration to evaluate a series that doesn't converge absolutely

Let $P(z)$ and $Q(z)$ be polynomials such that the degree of $Q(z)$ is exactly one degree more than the degree of $P(z)$. And assume that $ \displaystyle \sum_{n=-\infty}^{\infty} (-1)^{n} \frac{P(n)}...
12
votes
1answer
405 views

Closed form for $\sum_{n=-\infty}^\infty \frac{1}{(z+n)^2+a^2}$

I want to express $$\sum_{n=-\infty}^\infty \dfrac{1}{(z+n)^2+a^2}$$ in closed form. What comes to mind is the formula $$\pi\cot\pi z = \dfrac{1}{z}+\sum_{n\ne 0}\left(\dfrac{1}{z-n}+\dfrac1n\right)=\...
3
votes
2answers
140 views

Series sum $\sum 1/(n^2+(n+1)^2)$

In an exercise, I caculate the Fourier expansion of $e^x$ over $[0,\pi]$ is $$e^x\sim \frac{e^\pi-1}{\pi}+\frac{2(e^\pi-1)}{\pi}\sum_{n=1}^\infty \frac{\cos 2nx}{4n^2+1}+\frac{4(1-e^\pi)}{\pi}\sum_{n=...
2
votes
3answers
121 views

Methods for evaluating $\sum_{n=1}^\infty \frac1{a+(n-1)n}$

I am interested in methods for evaluating the sum $$\sum_{n=1}^\infty \frac1{a+(n-1)n}.$$ Indeed I will give my own answer below using the Residue Theorem. Please feel free to post other methods for ...
2
votes
1answer
169 views

Show that $\sum_{n=0}^{\infty}\frac{1}{4n^4+1}=\frac{1}{2}+\frac{\pi}{4}\tanh\left(\frac{\pi}{2}\right)$

Show that $$\sum_{n=0}^{\infty}\frac{1}{4n^4+1}=\frac{1}{2}+\frac{\pi}{4}\tanh\left(\frac{\pi}{2}\right)$$. I am thinking of using Fourier series and Parseval's identity to tackle this, I tried $x^...
5
votes
2answers
209 views

Summation of the series$\sum_{n=1}^\infty\frac{1}{n^2+4}$

Evaluate the sum of the following series $$\sum_{n=1}^\infty\frac{1}{n^2+4}$$ I saw a video in youtube where it is solved using complex analysis. What other method can be used to solve this?
0
votes
4answers
162 views

How to compute $\sum_{k = 1}^{\infty} \frac{1}{k^2-\frac{1}{16}}$? [duplicate]

How do I compute $\sum_{k = 1}^{\infty} \frac{1}{k^2-\frac{1}{16}}$ ? Mathematica says the sum converges and it somewhat looks like the Basel problem, but so far I do not know how to approach it.

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