Chapter 11 - 11.3 - Geometric Sequences and Series - 11.3 Exercises - Page 796: 72

$S=\frac{6}{5}$

$\displaystyle \sum_{n=0}^{∞}2(-\frac{2}{3})^n=2(-\frac{2}{3})^0+2(-\frac{2}{3})^1+2(-\frac{2}{3})^2+...$ $a_1=2(-\frac{2}{3})^0=2$ $r=\frac{a_2}{a_1}=\frac{2(-\frac{2}{3})^1}{2(-\frac{2}{3})^0}=-\frac{2}{3}$ $S=\frac{a_1}{1-r}=\frac{2}{1-(-\frac{2}{3})}=\frac{2}{\frac{5}{3}}=\frac{6}{5}$