Chapter 12 Sequences and Series - 12.4 Find Sums of Infinite Geometric Series - 12.4 Exercises - Skill Practice - Page 823: 21

$\dfrac{1}{2}$

Here, we have $a_n= a_1 r^{n-1}$ for the Geometric series. First term $a_1= \dfrac{2}{3}$ and Common ratio $r=\dfrac{-1}{3}$ The sum of an infinite Geometric Series can be found using: $S_n=\dfrac{a_1}{1-r}$ Thus, $S_n=\dfrac{ \dfrac{2}{3}}{1-( \dfrac{-1}{3})}=\dfrac{ \dfrac{2}{3}}{1+ \dfrac{1}{3}}$ Hence, $S_n=\dfrac{1}{2}$