Find the domain of convergence of $\sum_{n = 1}^{\infty} \frac{\cos(nz)}{n^2}$.
I'm not so smart - at first, I expanded the numerator in the sequence into $\Re(e^{niz})$, and tried to do an easy ratio test that indicated it diverged everywhere.
However, afterward, realizing this mistake, I expanded $\cos(nz)$ into $\frac{e^{niz}+e^{-niz}}{2}$ so have $\sum_{n = 1}^{\infty} \frac{e^{niz}+e^{-niz}}{2n^2}$. Where do I go from here?
If a series does converge then it's terms goes to zero, but for $z\not \in \mathbb{R}$ the term $\frac{e^{niz}+e^{-niz}}{2n^2}$ does not converge to zero, as can be seen by expressing $z=x+iy$ and noting that you always will have (in any case $y>0$ or $y<0$) an exponential that grows to infinity. Hence the series only converges for $z$ real.