Chapter 5.5, Problem 292E

For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false.

292. Suppose that $a_n$ is a sequence of positive real numbers and that ${\displaystyle \underset{n1}{\overset{\infty }{\sum }}{a}_n^{+}}$ converge. Suppose that $b_n$ is an arbitrary sequence of ones minus ones . Does ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{a}_n{b}_n}$ necessarily converge?