Chapter 9.6, Problem 50ES

Let ${\displaystyle \sum u_k}$ be a series and define series ${\displaystyle \sum p_k}$ and ${\displaystyle \sum q_k}$ so that $p_k=\left\{{}_{0,}^{u_k,}{}_{u_k\le 0}^{u_k>0}\text{and}{q}_k=\left\{{}_{-u_k,}^{0,}{}_{u_k<0}^{u_k\ge 0}\right.\right.$

(a) Show that ${\displaystyle \sum u_k}$ converges absolutely if and only if ${\displaystyle \sum p_k}$ and ${\displaystyle \sum q_k}$ both coverage.

(b) Show that if one of ${\displaystyle \sum p_k}\text{or}{\displaystyle \sum q_k}$ converges and the other diverges, then ${\displaystyle \sum u_k}$ diverges.

(c) Show that if ${\displaystyle \sum u_k}$ converges conditionally, then both ${\displaystyle \sum p_k}\text{or}{\displaystyle \sum q_k}$ diverge.