# Thread: Absolute convergence of complex series

1. ## Absolute convergence of complex series

Am I right in thinking a series converges absolutely if the modulus of the sum converges?

I have a question with two summations,

[Sum from n=1 to inf] ( i^n ) / n

[Sum from n=1 to inf] ( i^n ) / n^2

Asking which of these (if any) converge absolutely, with reasoning.

Intuitively they both converge absolutely, but the second one faster as the divisor would become greater quicker?

Could someone please help me?

2. Actually the first one doesn't.

If you take the modulus of all the terms you get:
$\sum_{i=1}^n \frac 1 n$
which is the time-honoured divergent harmonic series:
$\frac 1 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 +$
for which there is an elementary proof that it does in fact diverge, which can be found everywhere if you look.

So intuition is the loser here ...