I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. One way of doing it is to expand $K_2(nx)$ for small $x$ and then sum over $n$. However, doing this, we are implicity assuming $nx \ll 1$, while, on the other hand, $n$ goes all the way to infinity in the sum.
Is there a smart way of expanding this function?