[math-fun] Convergence question

We know the series for Zeta(s) converges for all s > 1. So, I'm wondering about functions e(n) of n that decrease to 1 slowly enough that A(e) = Sum_{n=1..oo} 1/n^e(n) converges. For example, what about letting e(n) = f_c(n) := 1 + 1/n^c ? Does A(f_c) converge for c close enough to 0, and if so, for which c ? (E.g., I tried using Mathematica to get an idea for c = 1/2, and was unable to tell after 30 minutes whether A(f_(1/2)) converges.) --Dan