Let $(a_n)_{n\in\mathbb{N}}\subseteq \mathbb{R}_{>0}$ be a convergent sequence. Is $$\sum\limits_{n=1}^\infty\frac{1}{n^2}(a_1+\cdots+ a_n)$$ convergent?
I don't know if this is correct or not and tried to find a counterexample so far, unsuccessfully, but I think it doesn't converge in general.
If one can choose $(a_n)_n$ as in this question Prove $a_1+\cdots+a_n=\dfrac{(a_1+a_n)n}{2}$ inductively. , then $\sum\limits_{n=1}^\infty\frac{1}{n^2}(a_1+\cdots+ a_n)$ does not converge due to the divergence of the harmonic series. But I would like to have a more concrete counterexample.
I am happy about any hint and help. Thank you