I'm asked to find an example of a diverging sequence $\sum_{n\in\mathbb{N}}a_n$ such that $\lim_{n\to\infty} a_n = 0$ but there exists parenthesization such that $\sum_{n\in\mathbb{N}}a'_n<\infty$ ($a'_n$ is the parenthesised sequence).
Clearly, we need to find a "forbidden" parenthesization that will change the convergence status of the sequence.
We learned that parenthesization is allowed in 2 cases:
- If the general term of the sequence $\to_{n\to\infty}{0}$ and the number of terms in each parentheses is bounded.
- All the terms in each parentheses are of the same sign.
Couldn't find one so far... Will appreciate your help.
Thanks.