Convergence of a series $sumlimits_{n=1}^inftyleft(frac{a_n}{n^p}right)^frac{1}{2}$

I've got a question about the convergence of a series during studying analysis.

If I know that a series of positive real numbers $$\sum_{n=1}^\infty a_n$$ converge, why does $$\sum_{n=1}^\infty\left(\frac{a_n}{n^p}\right)^\frac{1}{2}$$ also converge for $p>1$?

Although I know about many convergence tests, I don't know how to apply those tests for this case. Since this problem is the form of "series A converge → series B converge", I've been thinking that it must be verified by using some "comparison" tests. Is this thinking correct?

All advice is welcome^_^
Thanks.

Answer

Since
$$ab\leq a^2+b^2$$
we have
$$\sum_{n=1}^\infty\left(\frac{a_n}{n^p}\right)^\frac{1}{2}\leq \sum_{n=1}^\infty a_n+\sum_{n=1}^\infty\frac{1}{n^p} $$