<s>[INST] How to calculate $\sum_{k=1}^{\infty}\frac{1}{(a^2+k^2)^2}$ after calculating $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Parseval identity? The task is to calculate sum $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Fourier coefficients of $f(x)=e^{ax}$. First of all I calculated Fourier coefficients of the sum:
$$\hat{f(k)}=\frac{1}{2\pi}\int_{0}^{2\pi}e^{ax}e^{-ikx}dx=\frac{e^{a2\pi}-1}{2\pi(a-ik)}$$
Using Parseval identity I calculated the sum:
$$\frac{e^{4a\pi}-1}{2a}=\int_{0}^{2\pi}e^{2ax}dx=\sum_{k=-\infty}^{\infty}|\hat{f(k)}|^2=\frac{(e^{2a\pi}-1)^2}{4\pi^2}\cdot \sum_{k=-\infty}^{\infty}\frac{1}{a^2+k^2}=\frac{(e^{2a\pi}-1)^2}{4\pi^2}\cdot (2\cdot\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}-\frac{1}{a^2})$$
and thus:
$$\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}=\frac{a\pi(e^{2a\pi}+1)}{2a^2(e^{2a\pi}-1)}-\frac{1}{2a^2}$$
And in this task there is a subtask to calculate $\sum_{k=1}^{\infty}\frac{1}{(a^2+k^2)^2}$, so it seems it has to be calculated using previous sum, but I found no way to connect these sums, so what could be a the idea to deal with this?
Thanks in advance!
 [/INST] HINT
Note that
$$\dfrac{d}{da} \left(\dfrac1{a^2+k^2}\right) = - \dfrac{2a}{(a^2+k^2)^2}$$
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