Evaluate the sum $\sum_{k=1}^\infty \big(\frac{1}{36k^2-1}+\frac{2}{(36k^2-1)^2}\big)$

Question

I have to evaluate the series: $$\sum_{k=1}^\infty \left(\frac{1}{36k^2-1}+\frac{2}{(36k^2-1)^2}\right).$$

I tried using the identity $\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}$ but I got stuck.

Answer 1

A more general approach, proving the identity: $$ \sum_{k\geq 1}\frac{k^2+x^2}{(k^2-x^2)^2}=-\frac{1}{2x^2}+\frac{\pi^2}{2\sin^2(\pi x)}\tag{1}$$ for any $x\in(-1,1)$ is to check that both the LHS and the RHS of $(1)$ are meromorphic functions with the same-behaving singularities at any $x\in\mathbb{Z}\setminus\{0\}$. For instance, $(1)$ is just a consequence of: $$ \sum_{n\in\mathbb{Z}}\frac{1}{(z-n)^2}=\frac{\pi^2}{\sin^2(\pi z)}\tag{2} $$ since $\frac{1}{(k-x)^2}+\frac{1}{(k+x)^2} = 2\cdot\frac{k^2+x^2}{(k^2-x^2)^2}$. $(2)$ can be seen as a consequence of the reflection formula for the $\Gamma$ function, too, since: $$ \sum_{n\in\mathbb{Z}}\frac{1}{(z-n)^2}= \psi'(-z)+\psi'(1+z). \tag{3}$$ If we plug $x=\frac{1}{6}$ in $(1)$, we get:

$$ \sum_{k\geq 1}\frac{36k^2+1}{(36k^2-1)^2} = \color{red}{\frac{\pi^2-9}{18}}\tag{4} $$

and it is possible to derive such identity also through Fourier-analytic methods.

Still another way is to notice that our series is just $\frac{1}{2}\cdot L(\chi,2)$ where $\chi$ is the principal Dirichlet character $\!\!\pmod{6}$, hence our series depends on $\text{Li}_2$ evaluated at the sixth roots of unity.

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An elementary proof of $(4)$ comes from the inclusion-exclusion principle: $$ \sum_{k\geq 0}\left(\frac{1}{(6k+1)^2}+\frac{1}{(6k+5)^2}\right) = \sum_{\substack{n\geq 1 \\ 2\nmid n \\ 3\nmid n}}\frac{1}{n^2} = \sum_{n\geq 1}\frac{1}{n^2}-\sum_{\substack{n\geq 1 \\ 2\mid n}}\frac{1}{n^2}-\sum_{\substack{n\geq 1 \\ 3\mid n}}\frac{1}{n^2}+\sum_{\substack{n\geq 1 \\ 6\mid n}}\frac{1}{n^2}$$ gives:

$$ \sum_{k\geq 0}\left(\frac{1}{(6k+1)^2}+\frac{1}{(6k+5)^2}\right) = \left(1-\frac{1}{4}-\frac{1}{9}+\frac{1}{36}\right) = \frac{2}{3}\cdot\zeta(2) = \color{red}{\frac{\pi^2}{9}}. \tag{5}$$