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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

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  • $\begingroup$ Try to think of a simple sequence $a_n$ that satisfies the hypothesis. $\endgroup$ – user84413 Mar 2 '16 at 4:55
  • $\begingroup$ Your sentence "This is true only when ..." is totally false, so make sure you understand why you're wrong. $\endgroup$ – Ted Shifrin Mar 2 '16 at 5:43
  • $\begingroup$ You may also note that the converse of above statement is true. $\endgroup$ – Arpit Kansal Mar 2 '16 at 9:04
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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 $p=1$.

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    $\begingroup$ I was going to go with $a_n = (-1)^n$, but this works too. $\endgroup$ – Cameron Williams Mar 2 '16 at 4:52
  • $\begingroup$ 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, $\endgroup$ – Steven Stadnicki Mar 2 '16 at 5:19

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