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?