Question 7.27: Prove that ∑n=1^∞ 1/n² + a² = π/2a coth πa - 1/2a² where a &...

The result of Problem 7.26 can be written in the form

\sum\limits_{n=-\infty}^{-1} \frac{1}{n^{2}+a^{2}}+\frac{1}{a^{2}}+\sum\limits_{n=1}^{\infty} \frac{1}{n^{2}+a^{2}}=\frac{\pi}{a} \operatorname{coth} \pi a

or

2 \sum\limits_{n=1}^{\infty} \frac{1}{n^{2}+a^{2}}+\frac{1}{a^{2}}=\frac{\pi}{a} \operatorname{coth} \pi a

which gives the required result.