Finding $\sum_{k=0}^{\infty} \frac{1}{k^4+1}$ via residue theorem

Question

I want to find $\sum\limits_{k=0}^{\infty} \dfrac{1}{k^4+1}$ via residue theorem. Especially form of cotangent or tangent functions.

http://math.stackexchange.com/questions/384780/closed-form-for-sum-n-infty-infty-frac1n4a4/384839#384839 — Ron Gordon, Oct 30 '16 at 11:28
http://www.supermath.info/InfiniteSeriesandtheResidueTheorem.pdf — Marco Cantarini, Oct 30 '16 at 12:13
Possible duplicate of How do I calculate $\sum_{n\geq1}\frac{1}{n^4+1}$? — Martin R, Oct 30 '16 at 12:47

score 1 · Answer 1 · answered Oct 30 '16 at 15:19

1

Hint

$$\sum_{n\geq1}\frac{1}{n^4-a^4} = \frac{1}{2a^2}-\frac{\pi}{4 a^3}(\cot \pi a+\coth \pi a)$$

Hence for $a^4 = 1$...

answered Oct 30 '16 at 15:19

Enrico M.

1 Answers1