trigonometry - Sequences where $sumlimits_{n=k}^{infty}{a_n}=sumlimits_{n=k}^{infty}{a_n^2}$

I was recently looking at the series $\sum_{n=1}^{\infty}{\sin{n}\over{n}}$, for which the value quite cleanly comes out to be ${1\over2}(\pi-1)$, which is a rather cool closed form.

I then wondered what would happen to the value of the series if all the terms in the series were squared.

Turns out... nothing happens!

$\displaystyle\sum_{n=1}^{\infty}{\left({\sin{n}\over{n}}\right)}^2=\sum_{n=1}^{\infty}{\sin{n}\over{n}}={1\over2}(\pi-1)$.

This is a rather cool result, and I was wondering if there are any other simple series that share this property? Or, more generalized, series for which raising the terms to the power $m$ yields the same result as raising them to the power $p$.