Q:

Determine the vaues $p$ such that the following series converge?

Let $a_{n}\geq 0$ and $\sum_{1}^{\infty}a_{n}$ be a convergent series. Determine the values $p$ such that $\sum_{1}^{\infty}\frac{\sqrt{a_{n}}}{n^{p}}$ converge?

A:

Hint: $(1)$ Use the Schwarz Inequality, $x_i=\sqrt{a_i}$, $y_i=\frac{1}{i^p}$ to show that there is convergence if $p\gt \frac{1}{2}$. $(2)$ Show we cannot do better by using $a_n =\frac{1}{n\log^a n}$ for suitable $a$. 

