For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$

Question

For $a>0$,the series $\sum_{n=1}^\infty a^{\ln n}$ is convergent if and only if

(1). $0<a<e$

(2). $0<a\leq e$

(3). $0<a<\frac{1}{e}$

(4). $0<a\leq \frac{1}{e}$

I tried with $a=e$ as well as $a=\frac{1}{e}$. I can eliminate (1),(2) and (4), but how can I confirm (3)?

Answer 1

Note that$$a^{\log n}=e^{(\log n)(\log a)}=n^{\log a}$$and that $\sum_{n=1}^\infty n^\alpha$ converges if and only if $\alpha<-1$. So, your series converges if and only if $\log a<-1$, which is equivalent to $a<\frac1e$.