If $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty \frac{\sqrt a_n}{n^p}$ diverges, then p $\in$ {?} Let {$a_n$} be a sequence of non-negative real numbers such that the series $$\sum_{n=1}^\infty a_n$$ is convergent.
If p is a real number such that the series $$\sum_{n=1}^\infty \frac{\sqrt a_n}{n^p}$$
diverges, then what can be said about the value of p?

Seems like we'll have to make use of the A.M. $\ge$ G.M. inequality here, so that we have $$\frac {\sqrt{a_n}} {n^p} \leq a_n + \frac{1}{n^{2p}}$$
Now, this diverges for p $\le$ $\frac12$. So that must be the required answer.
Can anyone show how to do this using Cauchy-Schwarz inequality? It is simple I reckon.:)
