Series with complex terms, convergence Could you tell me how to determine convergence of series with terms being products of real and complex numbers, like this:
$\sum_{n=1} ^{\infty}\frac{n (2+i)^n}{2^n}$ , $ \ \ \ \ \sum_{n=1} ^{\infty}\frac{1}{\sqrt{n} +i}$?
I know that $\sum (a_n +ib_n)$ is convergent iff $\sum a_n$ converges and $\sum b_n$ converges.
(How) can I use it here?  

Just like for series of reals (the underlying theorems are in the complex numbers, after all).
Your first series doesn't converge. By the ratio test, as $n \rightarrow \infty$:
$$
\lvert \frac{(n + 1) (2 + i)^{n + 1}/2^{n + 1}}{n (2 + i)^n / 2^n} \rvert = \lvert \frac{(n + 1) (2 + i)}{2 n} \rvert =
\frac{(n + 1) \sqrt{5}}{2 n} \rightarrow \frac{\sqrt{5}}{2}
$$
