If Series $\sum \sqrt\frac{a_n}{n}$ converges then $\sum a_n$ converges?

Question

Given $a_n>0$. If Series $\sum \sqrt\frac{a_n}{n}$ converges then $\sum a_n$ converges?

My approach:

Ihad used atmost every method like 1)AM and GM inequality 2) cauchy Swartz inequality 3) by definition of convergent series and so on.. I am not able to prove this statement. I am thinking now this statement is false, neither I am able to find counter example for this. Any Hint/direction please..

If $a_n$ decreases here is a postingSuppose $\sum_{n=1}^{\infty}\sqrt{{a_{n}}/{n}}$ is convergent. Prove that $\sum_{n=1}^{\infty}a_{n}$ is also convergent. — Mittens, May 24 '22 at 17:36

score 11 · Accepted Answer · answered May 24 '22 at 17:48

11

Counterexample: $a_n=1/k$ if $n=k^2$ with $k \in \mathbf{N}$, and $a_n=0$ otherwise. In this case, $\sqrt{a_n/n} = 1/k^{3/2}$ if $n=k^2$ with $k \in \mathbf{N}$, and $\sqrt{a_n/n} = 0$ otherwise.

answered May 24 '22 at 17:48

Christophe Leuridan

8,223

Thankyou so much.. – Tony May 24 '22 at 17:56
2

+1. My counter-example is $a_n=4^{-n}$ if $n$ is not a power of $4$, and $a_n=1$ if $n $ is a power of $4$. BTW the Q calls for $a_n>0$, unlike your A, but this is easily remedied. – DanielWainfleet May 24 '22 at 18:16

If Series $\sum \sqrt\frac{a_n}{n}$ converges then $\sum a_n$ converges?

1 Answers1