Chapter 8.4, Problem 58PS

To determine

To show: that the series ${\displaystyle \sum a_k^2}$ converges provided $0<a_k<1$ for all $k$ and ${\displaystyle \sum a_k}$ converges. Also find a counter example where ${\displaystyle \sum a_k^2}$ converges and ${\displaystyle \sum a_k}$ diverges.