Chapter 11.6, Problem 53E

Suppose the series ${\displaystyle \sum a_n}$ is conditionally convergent.

(a) Prove that the series ${\displaystyle \sum n^2{a}_n}$ is divergent.

(b) Conditional convergence of ${\displaystyle \sum a_n}$ is not enough to determine whether ${\displaystyle \sum n{a}_n}$ is convergent. Show this by giving an example of a conditionally convergent series such that ${\displaystyle \sum n{a}_n}$ converges and an example where ${\displaystyle \sum n{a}_n}$ diverges.