$1+\frac{1}{(N-1)²}+\frac{1}{(N+1)²}+\frac{1}{(2N-1)²}+\frac{1}{(2N+1)²}+...$ series

Question

I would like to know if there are references on the integral

$1+\frac{1}{(N-1)²}+\frac{1}{(N+1)²}+\frac{1}{(2N-1)²}+\frac{1}{(2N+1)²}+...$ N is a natural number greatest 2

Answer 1

$$ \begin{align} \color{red}{S}&=1+\sum_{k=1}^{\infty}\frac{1}{(kN-1)^2}\,+\,\sum_{k=1}^{\infty}\frac{1}{(kN+1)^2} \\[2mm] &=1+\frac{1}{N^2}\sum_{k=1}^{\infty}\frac{1}{(k-1/N)^2}\,+\,\frac{1}{N^2}\sum_{k=1}^{\infty}\frac{1}{(k+1/N)^2} \\[2mm] &=1+\frac{1}{N^2}\sum_{k=1}^{\infty}\frac{1}{(k\color{red}{-1+1}-1/N)^2}\,+\,\frac{1}{N^2}\sum_{k=1}^{\infty}\frac{1}{(k\color{red}{-1+1}+1/N)^2} \\[2mm] &=1+\frac{1}{N^2}\sum_{k=\color{red}{0}}^{\infty}\frac{1}{(k+(1-1/N))^2}\,+\,\frac{1}{N^2}\sum_{k=\color{red}{0}}^{\infty}\frac{1}{(k+(1+1/N))^2} \\[2mm] &=\color{red}{1+\frac{1}{N^2}\left[\,\zeta\left(2,1-\frac{1}{N}\right)+\zeta\left(2,1+\frac{1}{N}\right)\,\right]} \end{align} $$ Hurwitz Zeta function

Answer 2

Similar to Hazem Orabi's answer.

$${S}=1+\sum_{k=1}^{\infty}\frac{1}{(kN-1)^2}\,+\,\sum_{k=1}^{\infty}\frac{1}{(kN+1)^2} $$ $$S=1+\frac{\psi ^{(1)}\left(1-\frac{1}{N}\right)}{N^2}+\frac{\psi ^{(1)}\left(1+\frac{1}{N}\right)}{N^2}=\color{blue}{\frac{\pi ^2 }{N^2}\csc ^2\left(\frac{\pi }{N}\right)}$$

from the identity $$\psi ^{(1)}(1-a)+\psi ^{(1)}(1+a)=\pi ^2 \csc ^2(\pi a)-\frac{1}{a^2}$$