Wsrtjtyk

I have to give an example of a convergent series $sum a_{n}$ for which $sum a_{n}^2 $ diverges.

I think that such a series cannot exist because if $sum a_{n}$ converges absolutely then $sum a_{n}^2 $ will always converge right?

The alternating series test gives a wealth of examples. Take
$$
a_n=(-1)^n/sqrt{n}
$$
for example.

Try $$sum_{n=1}^infty frac{(-1)^n}{sqrt{n}}$$

More generally,
if $a_n = dfrac{(-1)^n}{n^{1/(2m)}}$,
then
$sum_{n=1}^{infty} a_n
$
converges for integer $m ge 0$
and
$sum_{n=1}^{infty} a_n^{2m}
$
diverges.

The simplest example I have in mind is :
$$sum_{n=1}^{+infty}frac{(-1)^n}{sqrt{n}} $$

Beware that convergent (CV) and absolutely convergent (ACV) can be very different.
Indeed, if $sum_{n=1}^{+infty} a_n $ is ACV then, $sum_{n=1}^{+infty} a_n^2$ is also ACV. To prove it, you can simply notice that, since $lim a_n=0$ then for $n$ big enough, $|a_n^2|<|a_n|$.

You can also ask yourself a more general question :
What are the function $f:mathbb R rightarrow mathbb R$ such that for all
$sum a_n$ CV (resp. ACV), $sum f(a_n)$ is CV (resp. ACV).

It is a (difficult) exercise to show that for $f:mathbb Rrightarrow mathbb R$,

$$sum a_n ~CV Rightarrow sum f(a_n) ~CV quad text{iff} quad exists eta>0,exists lambdain mathbb R,forall xin ]-eta,eta[, quad f(x)=lambda x $$

$$sum a_n ~CV Rightarrow sum f(a_n) ~ACV quad text{iff} quad exists eta>0,forall xin ]-eta,eta[, quad f(x)=0 quadquadquad~~$$

$$sum a_n ~ACV Rightarrow sum f(a_n) ~ACV quad text{iff} quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad$$ $$quad f(0)=0 text{ and }exists eta>0,exists M >0,forall xin ]-eta,eta[, |f(x)|leq M |x| $$

$$sum a_n ACV Rightarrow sum f(a_n) ~CV quad text{iff} quad sum a_n ~ACV Rightarrow sum f(a_n) ~ACV~~~~~~~~~~~~~~ $$