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# self learning - necessary and sufficient condition for convergence of power series

Show that $$\sum_{n=1}^{\infty} a^{\ln~n}$$ is convergent if and only if $$0.

Proof:

$$\ln~n for all $$n \geq 1$$.Hence the necessary condition for the series to be convergent will be $$\lim_{n \rightarrow \infty} a^{\ln~n}=0$$. And this will happen if $$0. Please give me some hint about how to proceed further.

For $$0, $$a^{\ln(n)}$$ is a decreasing sequence. Hence by Cauchy condensation test, $$\sum a^{\ln(n)}\text{ is convergent} \iff \sum 2^na^{\ln(2^n)}\text{ is convergent}$$ The last sum is a geometric series, convergent for $$0.
Another approach would be to see that $$a^{\ln(n)}=n^{\ln(a)}$$, $$\sum n^{\ln(a)}$$ is a Dirichlet series, convergent only for $$\ln(a)<-1$$.