How to show that $\sum\limits_{n=1}^\infty \exp(i\,nz)$ converges? How do I show that $$ f(z) = \exp(i\, z) + \exp(i\, 2z) + \ldots + \exp(i\, nz) + \ldots $$ converges? Problem is taken from a Yahoo! Answers question: "Find the infinite sum of sin(n)/n?".

It looks like a geometric series to me. Recall that a geometric series is of the form
$S = a + ar + ar^2 + ar^3 + \cdots \, .$
In this case, the first term $a = e^{iz}$ while the common ratio $r = e^{iz}.$
$S = \frac{a}{1-r} = \frac{e^{iz}}{1-e^{iz}} \, . $
This function defines the analytic continuation of $f$ provided $e^{iz} \neq 1$.
