a sequence $\{s_n\}$ with $\sum s_n$ convergent

what would be $\{s_n\ge0\}$ such that $$\sum_{n=1}^\infty s_n$$ converges

but

$$\lim_{n\to\infty}(n s_n) \neq 0$$

$a_n=\frac{1}{n}$ if $n$ is a perfect square and $a_n=\frac{1}{n^2}$ otherwise.

P.S. This example is generic in the following sense:

Lemma If $a_n \geq 0, \sum a_n$ is convergent and $\lim_n na_n$ exists then $\lim na_n=0$.

Proof: Assume by contradiction that $\lim_n na_n \neq 0$. Then there exists some $M>0$ so that $na_n >M$ for all $n>N$. Then

$$a_n >M \frac{1}{n} \forall n >N \Rightarrow \sum_a_n =\infty$$ since the harmonic series diverges.

Thus, the only way you can construct an example as you want is by trying to make $\lim_n na_n$ not to exists.

An counter-example inlovning positive and negative terms is when

$a_n \frac{(-1)^n}{\sqrt{n}}$ the convegence can be check using the Alternating series test.

For an example with positive terms choose

$a_n= \frac{1}{n}$ when $n$ is perfect square otherwise let it be $0$