Question 7.31: Suppose a≠0, ±1, ±2, ... . Prove that a² + 1/(a² - 1)² - a² ...
Suppose a \neq 0, \pm 1, \pm 2, \ldots Prove that
\frac{a^{2}+1}{\left(a^{2}-1\right)^{2}}-\frac{a^{2}+4}{\left(a^{2}-4\right)^{2}}+\frac{a^{2}+9}{\left(a^{2}-9\right)^{2}}-\cdots=\frac{1}{2 a^{2}}-\frac{\pi^{2} \cos \pi a}{2 \sin ^{2} \pi a}
The result of Problem 7.30 can be written in the form
\frac{1}{a^{2}}-\left\{\frac{1}{(a+1)^{2}}+\frac{1}{(a-1)^{2}}\right\}+\left\{\frac{1}{(a+2)^{2}}+\frac{1}{(a-2)^{2}}\right\}+\cdots=\frac{\pi^{2} \cos \pi a}{\sin ^{2} \pi a}
or
\frac{1}{a^{2}}-\frac{2\left(a^{2}+1\right)}{\left(a^{2}-1\right)^{2}}+\frac{2\left(a^{2}+4\right)}{\left(a^{2}-4\right)^{2}}-\frac{2\left(a^{2}+9\right)}{\left(a^{2}-9\right)^{2}}+\cdots=\frac{\pi^{2} \cos \pi a}{\sin ^{2} \pi a}
from which the required result follows. Note that the grouping of terms in the infinite series is permissible since the series is absolutely convergent.