In case anyone else read this, I have found the solution on the internet. It was embarassingly simple. By the AM-GM inequality:
a_n + \frac{1}{n^2} \ge 2\frac{\sqrt{a_n}}{n}.
The left hand series converges, so by direct comparison, the right hand series also converges.
Our instructor assigned a problem from Rudin's Principles of Mathematical Analysis; a problem which I have been unable to solve after giving it good thought.
The statement is:
"Prove that the convergence of SUM[an] implies the convergence of SUM[sqrt(an)/n], if an >= 0."
The instructor...