Series of sequence with terms over series.

The series $\sum a_k$ converges if and only if the sequence of partial sums $s_k$ converges.

Suppose $\sum a_k$ converges where $a_k>0$. Then $\lim s_k >0 $ exists. Apply the limit comparison test to $b_k/a_k=s_k^{-1}.$ Hence, if $\sum a_k$ converges then so does $\sum b_k.$ If $\sum b_k$ diverges then $\sum a_k$ diverges.

Suppose, on the other hand, that $\sum a_k$ diverges with $a_k> 0$. Then $\lim s_k= \infty.$ For $m > n$, since $s_k$ is increasing

$$\sum_{k=n+1}^{m}b_k > \frac1{s_m}\sum_{k=n+1}^{m}a_k =1 - \frac{s_n}{s_m},$$

and for fixed $n$, the RHS converges to $1$ as $m \rightarrow \infty$. Hence, for $m$ sufficiently large

$$\sum_{k=n+1}^{m}b_k > \frac1{2},$$

and the Cauchy criterion fails -- so $\sum b_k$ also diverges. If $\sum b_k$ converges then $\sum a_k$ converges.

Therefore, $\sum a_k$ and $\sum b_k$ either both converge or both diverge.