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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..

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1 Answer 1

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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.

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  • $\begingroup$ Thankyou so much.. $\endgroup$
    – Tony
    May 24, 2022 at 17:56
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    $\begingroup$ +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. $\endgroup$
    – DanielWainfleet
    May 24, 2022 at 18:16

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