real analysis - examples of conditionally convergent series other than alternating harmonic series

I've been trying to find interesting examples of conditionally convergent series but have been unsuccessful. I'd particularly like to find a conditionally convergent series of the following form:

$\sum_{n=1}^\infty a_n$ where $a_n=f(n,z)$ with $Im(z)≠0$

(or even better $a_n=f(n,z^n)$, with $Im(z)≠0$)

but if you know of any interesting conditionally convergent series at all, please share. Alternating harmonic series appears to be the most popular example everywhere.