Range of values for convergence of a series

Question

Question: For $a>0$. the series $$ \sum_{n=2}^{\infty} a^{\log_e n}$$

is convergent for which range of values of $a$?

My Approach:

The divergence test of checking $\lim_{n\to\infty} t_n =0 $ gives $0<a<1$

So does the ratio test and the root test.

I am fairly new to this topic, so I am not sure how to proceed.

Edit: The answer is $0<a<1/e$ I fail to see how they arrive at this.

Answer 1

Hint:

This is indeed a Riemann series, since you can rewrite its general term as $$a^{\log n}=\mathrm e^{\log n\log a}=n^{\log a}.$$ As $\sum_n n^\alpha$ converges if and only if $\alpha <-1$, this yields $$\log a<-1\iff 0<a<\dfrac1{\mathrm{e}}.$$