# If $\sum_{n=1}^{\infty} a_n^{2}$ converges, then so does $\sum_{n=1}^{\infty} \frac {a_n}{n}$

Let $$a_1,a_2,a_3,\ldots$$ be reals. Prove that if $$\sum_{n=1}^{\infty} a_n^{2}$$ converges, then so 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.

• 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
• can you explain the transition for which ${a_n}^2 \leq \frac{|a_n|}{n} \Longrightarrow \frac{|a_n|}{n} \leq \frac{1}{n^2} + {a_n}^2$ ? – Jneven Jan 23 '19 at 14:41
• @Jneven $a_n^2 = |a_n||a_n|$, thus it follows that $a_n^2 \leq \frac{|a_n|}{n} \implies |a_n| \leq \frac{1}{n} \implies \frac{|a_n|}{n} \leq \frac{1}{n^2}$. You can add anything you'd like to the right side as long as it is nonnegative and the inequality will still hold. – cwk Jan 28 '19 at 19:37

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 $b_n=\frac 1n$.

• Many thanks really appreciate it. – user24930 Feb 24 '12 at 19:35
• Instead of $\sum {a_n}^2$, Let $a_n$ positive terms. if $\sum_{n=1}^\infty (a_n)^{3/2}<\infty$ then will $\sum_{n=1}^\infty a_n/n$ converges? – user464147 May 3 '19 at 16:16

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 $\frac{\pi^2}{6} = \sum_{n=1}^{\infty} \frac{1}{n^2}.$

• Many thanks really appreciate it. – user24930 Feb 24 '12 at 19:35
• Great answer :) – MathMan 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.

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

AM-GM:

$$b_n := a_n^2+ 1/n^2 \ge 2|a_n|(1/n)$$.

$$\sum b_n$$ converges.

$$S_n := 2\sum_{k=1}^{n}|a_n|(1/n)$$ is increasing and bounded above, hence converges, which implies

$$\sum a_n/n$$ converges.

Note that for every $$n\in\mathbb{N}$$ we have

$$\left(|a_n|-\frac{1}{n}\right)^2\geq0\Longrightarrow 2\frac{|a_n|}{n}\leq a_n^2+\frac{1}{n^2}.$$ Then we can use the comparison test to assert the convergence of $$\sum\dfrac{|a_n|}{n}$$. Once again, by the same test, we have that $$\sum\dfrac{a_n}{n}$$.

• You have added nothing new to already existing answers – Shailesh Nov 8 '19 at 1:56

Lets split the sum $$\sum_{n=1}^\infty a_n^2$$ into two sums (we can since it converges absolutely) , one is the sum of all $$a_n^2$$ where $$a_n\leq \frac1n$$ and the other of all $$a_n^2$$ such that $$a_n>\frac1n$$, now for the first sum we have $$\frac{a_n}n\leq \frac1{n^2}$$ and for second we have $$a_n^2=a_n\cdot a_n>a_n\cdot\frac1n$$ so we have that the corresponding two sums of $$\frac{a_n} n$$ also converge so we can combine them and rearrange (because both converge absolutely) them into $$\sum_{n=1}^\infty\frac{a_n}n$$