TITLE: Differentiability of $s=\sum\limits_{-\infty}^{\infty} {1\over (x-n)^2}$
QUESTION [5 upvotes]: How do I show that $s=\sum\limits_{-\infty}^{\infty} {1\over (x-n)^2}$ on $x\not\in \mathbb Z$ is differentiable without using its compact form? I realize that the sequence of sums $s_a=\sum\limits_{-a}^{a} {1\over (x-n)^2}$ is not uniformly convergent. 
I also tried to prove that it is continuous by using the usual $\varepsilon \over 3$ method. And it seems to apply because each $s_a$ is continuous and they converge pointwise to $s$. But then I realized that this should not be the right proof because I didn't use any special property of the given functions and the general case only works for uniform convergence. I am very confused. Please help!
Actually I do realize that if I could prove differentiability, continuity follows.
Thanks.

REPLY [5 votes]: OK, just to be exotic (?), let's see if we if we can get this from Morera's theorem.  Let $C$ be a simple closed curve neither winds around any integer nor passes through any integer.  Then
$$
\int\limits_C \sum_{n=-\infty}^\infty \frac{1}{(x-n)^2}\;dx = \sum_{n=-\infty}^\infty\  \int\limits_C  \frac{1}{(x-n)^2}\;dx = \sum_{n=-\infty}^\infty 0 = 0.
$$
The first equality follows from Fubini's theorem.*
The second equality follows from the fact that $C$ does not wind around any point where the holomorphic function $x\mapsto 1/(x-n)^2$ behaves badly (fails to be holomorphic).
Morera's theorem says that if the integral of a function along every simple closed curve that does not wind around any point not in the domain is $0$, then the function is holomorphic.
* Later note: Are the hypotheses of Fubini's theorem satisfied?  The terms of the sum are nonnegative and the sum converges to a finite number.  If one can show that it depends continuously on $x$, then the integral is that of a continuous function on a compact set, so that is also finite.
However, it now occurs to me that we don't need to go into that, because nonnegativity means we can cite Tonelli's theorem instead.  That says that one can interchange the order of two Lebesgue integrations for functions that are everywhere nonnegative, regardless of whether the value of the integral is finite or infinite.