# If series $\sum_n^{\infty} \frac{a_n}{n}$ converges, then does $\sum_n^{\infty} {a_n}^2$ converge?

If series $\sum_n^{\infty} \frac{a_n}{n}$ converges, then does $\sum_n^{\infty} {a_n}^2$ converge? If not give a counterexample.

This is true only when ${a_n}^2 ≤ \frac{|a_n|}{n}$ but I can't think of a counterexample

• Try to think of a simple sequence $a_n$ that satisfies the hypothesis. – user84413 Mar 2 '16 at 4:55
• Your sentence "This is true only when ..." is totally false, so make sure you understand why you're wrong. – Ted Shifrin Mar 2 '16 at 5:43
• You may also note that the converse of above statement is true. – Arpit Kansal Mar 2 '16 at 9:04

Counterexample: $a_n=\frac{1}{\sqrt{n}}$
This is from the following: $$\sum_n^{\infty} \frac{1}{n^p}$$ This converges when $p>1$, but diverges when $p=1$.
• I was going to go with $a_n = (-1)^n$, but this works too. – Cameron Williams Mar 2 '16 at 4:52
• I like this example better (and this is the one I was thinking of) because it shows that even absolute convergence isn't enough to force the result. In general, the $p$-series is an excellent starting point for trying to find counterexamples for this sort of result - if a conjecture is true for all the $p$-series then it may be worth trying to find a proof, – Steven Stadnicki Mar 2 '16 at 5:19