Convergence of series using ratio test

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Homework Statement

assume summation of series An converges with all An>0. Prove summation of sqrt(An)/n converges

Homework Equations

The Attempt at a Solution

I Tried using the ratio test which says if lim as n goes to infinity of |Bn+1/Bn|<1 then summation of Bn converges. I let Bn be Sqrt(An)/n and we have...

lim as n goes to infinity of |(sqrt(An+1/An)*n/n+1|. limit of n/n+1 goes to one so i need to prove that |sqrt(An+1/An)|<1. But I got stuck because just because An converges does not mean that |(An+1/An)|<1. Can someone help me or suggest on a different convergence test that I should use?

 

[Mark44]

For your series,
$$\lim_{n \to \infty} \sqrt{\frac{a_{n+1}}{a_n}} = 1$$

so the Ratio Test is not going to be any help.

What other tests do you know?