Chapter 11 - Section 11.2 - Series - 11.2 Exercises - Page 718: 81

The step $=1+(-1+1)+(-1+1)+(-1+1)+...$ is incorrect since addition is only associative for finitely many terms, not for infinitely many terms.

$0=0+0+0+...$ $=(1-1)+(1-1)+(1-1)+(1-1)+...$ $=1-1+1-1+1-1+...$ $=1+(-1+1)+(-1+1)+(-1+1)+...$ (1) $=1+0+0+0...=1$ Hence, the step $=1+(-1+1)+(-1+1)+(-1+1)+...$ is incorrect since addition is only associative for finitely many terms, not for infinitely many terms.