Converging series question, Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then does $\sum_{n=1}^{\infty} \frac {a_n}{n}$

Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then does $\sum_{n=1}^{\infty} \frac {a_n}{n}$

For this I have shown the case for when $a_n^{2} \le\frac {|a_n|}{n}$ $\Rightarrow$ $|a_n|\le\frac {1}{n}$ $\Rightarrow$ $\frac {|a_n|}{n} \le \frac{1}{n^{2}}$ and we know that $\sum_{n=1}^{\infty} \frac {1}{n^{2}}$ converges and hence $\sum_{n=1}^{\infty}\frac {a_n}{n}$ converges by the comparison test. Now considering $a_n^{2} \ge\frac {|a_n|}{n}$ $\Rightarrow$ $\frac {|a_n|}{n} \le a_n^{2}$ $\rightarrow$ combining the two cases for any n we have: $\frac {|a_n|}{n}\le\frac{1}{n^{2}}+a_n^{2}$ Hence using the comparion test again we know that $\sum_{n=1}^{\infty} a_n^{2}$ converges and $\sum_{n=1}^{\infty} \frac {1}{n^{2}}$ converges hence the sum converges so we can conclude that $\sum_{n=1}^{\infty} \frac {a_n}{n}$ is absoluetly convergent $\Rightarrow$ convergent. Not to sure if this is correct, any help would be much appreciated, many thanks.

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Great question! It is easy to check that this series converges since you know $\ell^2$ is Hilbert. But it is a very hard insight when you think only using the basic real Analysis. – checkmath Feb 23 '12 at 22:14
Many thanks I really appreciate the help. – user24930 Feb 24 '12 at 19:34

What you did is correct; in fact you can show that if $\{a_n\}$ and $\{b_n\}$ are two sequences of real numbers and $\sum_{n\geq 0}a_n^2$ and $\sum_{n\geq 0}b_n^2$ are convergent then the series $\sum_{n=0}^{+\infty}|a_nb_n|$ is convergent, noting that $0\leq |a_nb_n|\leq \max(a_n^2,b_n^2)\leq a_n^2+b_n^2$.

Your particular case is $b_n=\frac 1n$.

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Many thanks really appreciate it. – user24930 Feb 24 '12 at 19:35

Another approach is to note that for any positive integer $N,$ we have $\sum_{n=1}^{N} \frac{|a_n|}{n} \leq \sqrt{ \sum_{n= 1}^{N} a_{n}^{2}} \sqrt{ \sum_{n=1}^{N} \frac{1}{n^2}}$, and this is in turn less than $\frac{\pi}{\sqrt{6}}\sqrt{ \sum_{n= 1}^{N} a_{n}^{2} }.$ The first inequality follows by the Cauchy-Schwarz inequality, and the second follows by Euler's formula $\frac{\pi^2}{6} = \sum_{n=1}^{\infty} \frac{1}{n^2}.$

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Many thanks really appreciate it. – user24930 Feb 24 '12 at 19:35
Great answer :) – Wanderer Jan 27 '15 at 16:18

You are right.. you can also simplify things further using that $|ab| \leq {a^2 + b^2\over 2}$ for any $a$ and $b$, so that $|{a_n \over n}| \leq {a_n^2 \over 2} + {1 \over 2n^2}$ and thus your series converges absolutely as you are saying.

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Many thanks really appreciate it. – user24930 Feb 24 '12 at 19:35