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?".

For the series to converge, you have the following condition,
$$ |{\rm e}^{iz}| < 1 \,.$$
Assuming $z=x+iy$, we have,
$$ \left|{\rm e}^{i(x+iy)} \right| =   \left|{\rm e}^{ix-y)} \right| =  {\rm e}^{-y} < 1 \Rightarrow  -y < 0 \Rightarrow y > 0 \,.$$ 
