Test for convergence $\sum_{n\ge1}a^{1+1/2+\cdots+1/n}, a>0$ Test for convergence 
$$\sum_{n\ge1}a^{1+1/2+\cdots+1/n}, a>0$$
What would you suggest me to do here? If you have some nice proofs and want
to share them then I'll reward them with upvotes.  Thanks!

$1 + 1/2 + \ldots + 1/n = \ln(n) +O(1)$, and $a^{\ln(n)} = e^{\ln(a) \ln(n)} = n^{\ln(a)}$.
So it will converge for $\ln(a) < -1$ (i.e. $a < 1/e$) and diverge for $a \ge 1/e$.
