= Solution
Separate the $n=0$ term and pair $n$ with $-n$. For small $\alpha$,
$$
\frac{(-1)^n}{(n-\alpha)^2}+\frac{(-1)^n}{(n+\alpha)^2}=\frac{2(-1)^n}{n^2}+\frac{6\alpha^2(-1)^n}{n^4}+O(\alpha^4/n^6).
$$
Compare the constant and quadratic coefficients with the supplied expansion after subtracting $\alpha^{-2}$. It follows that
$$
\boxed{\sum_{n\geq1}\frac{(-1)^n}{n^2}=-\frac{\pi^2}{12},\qquad\sum_{n\geq1}\frac{(-1)^n}{n^4}=-\frac{7\pi^4}{720}.}
$$
The paired expansion is uniformly summable on any sufficiently small <closed interval> of $\alpha$, so coefficient comparison is legitimate.
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