Sequence VSS Series

Question

Given the limit to the infinity of sequence $A_n = 0$. Does the series $\Sigma_{n=1}^{\infty} A_n^2$ have to be convergent or divergent? If it is one of them, prove it. If it is either, give an example.

Answer 1

I change the notations of you and show the sequence with $a_n$ and series with $S=\Sigma (a_n)^2$.

If $$\lim_{n\to\infty} a_n=0$$

then

$$\Sigma a_n^2$$

can be either convergence or divergence.

Convergence example:

$a_n=\frac1n , ~~~~~~~~~~~~ S=\Sigma a_n^2=1+\frac14+\frac19+\dots=\text{convergent}$

Divergence example:

$a_n=\frac1{\sqrt n} , ~~~~~~~~~~~~ S=\Sigma a_n^2=1+\frac12+\frac13+\dots=\text{divergent}$