If $\sum\limits_{n=1}^\infty a_n $converges does it imply that $\sum\limits_{n=1}^\infty \dfrac {a_n^{1/4}}{n^{4/5}}$ is convergent?

Question

Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms, then is it true that $$\sum_{n=1}^\infty \dfrac {a_n^{1/4}}{n^{4/5}}$$ is convergent ?

I tried comparing with $\sum a_n$ , but without any progress.

Please help with any hint.

Answer 1

By Holder inequality: Proving Holder's inequality for sums with $ \frac14+ \frac{1}{\frac43}= 1$ we have, $$\sum_{n=1}^\infty \dfrac {a_n^{1/4}}{n^{4/5}} \le \left(\sum_{n=1}^\infty a_n\right)^{1/4}\left(\sum_{n=1}^\infty \dfrac {1}{n^{\frac{16}{15}}}\right)^{3/4}<\infty$$

Given that by Riemann series we have, $$\sum_{n=1}^\infty \dfrac {1}{n^{\frac{16}{15}}} =\sum_{n=1}^\infty \dfrac {1}{n^{1+\frac{1}{15}}}<\infty.$$