Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$

Asked Feb 20 '13 at 20:52

Active Feb 20 '13 at 23:24

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Being inspired by this post, I've wondered if the infinite series below may be expressed as
an intregral. I'm very curious about that.

$$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$

I'd appreciate your feedback here. How should I ponder over this case as a simple student?
Where should I start from and what techniques I need to think of? Thanks!

edited Apr 13 '17 at 12:21

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asked Feb 20 '13 at 20:52

user 1591719

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I finally solve it :) – Cortizol Feb 20 '13 at 21:21
@Cortizol: really??? – user 1591719 Feb 20 '13 at 21:21
I was thinking on my problem. I add answer (in my topic). – Cortizol Feb 20 '13 at 21:22
I'd also like to see if there is solution via integrals :) – Cortizol Feb 20 '13 at 21:30

Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$

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