Given a sum such as $\sum\_{n = -\infty}^\infty \frac{\lvert x \rvert}{(1 + n^2 x^2)^\frac{3}{2}}$, the first derivative (with respect to $x$) of the summand $\frac{\lvert x \rvert}{(1 + n^2 x^2)^\frac{3}{2}}$ does not exist. Nonetheless, numerical estimates such as:
[![numerical estimate](https://i.stack.imgur.com/UZNgp.png)](https://i.stack.imgur.com/UZNgp.png)
seem to indicate that the sum itself is a smooth function of $x$ (and it seems like all derivatives vanish at $x=0$).

What tools are there to show that this sum defines a smooth function of $x$, and to compute the derivatives (especially at $x = 0$)?