If $(a_n)$ be a sequence of positive real numbers such that $\sum a_n$ converges, then show that $\sum a_n^{\frac{n}{n+1}}$ converges as well.
Attempt:
We try to apply the Limit form of the Comparison Test here.
We note, $\lim_{n \to \infty} (a_n)^{\frac{n}{n+1}} =a_n$
Now, comparing the two series, $\displaystyle\lim_{n \to \infty} \frac{(a_n)^{\frac{n}{n+1}}}{a_n}=\lim_{n \to \infty} \frac{a_n}{a_n}=1$.
Therefore, if $\sum a_n$ converges, then $\sum a^{n/(n+1)}$ converges as well.
I have a feeling that I have made a mistake somewhere. I am doubtful about the application of the comparison test in this case.
Can anyone kindly correct/verify it?
Duplicate here.