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If $\sum $$a_{n}$ is convergent, prove $\sum \frac{a_n}{n}$ is also convergent.

I would say that $a_{n}\geq \frac{a_{n}}{n}$ so being $a_{n}$ convergent, $\frac{a_n}{n}$ also has to be convergent.

However, I think it is not enough...

5xum
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  • It is not: if $a_n=-1$, then $-1>-1/n$ is not true for $n>1$. If you know further that $a_n\geq 0$ for every $n$, then you can use Direct Comparison though – Hayden Jun 16 '14 at 10:25
  • It is sufficient if $a_n$ is a series with nonnegative terms. – Dario Jun 16 '14 at 10:25
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    An almost direct comparison is also possible if $\sum a_n$ converges absolutely (a strengthening of "converges with only positive terms"). – Arthur Jun 16 '14 at 10:28

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