## Thomas' Calculus 13th Edition

$\dfrac{3}{2}$
Formula to calculate the sum of a geometric series is: $S=\dfrac{a}{1-r}$; Since, we have two series $\sum_{n =0}^{ \infty} (\dfrac{2}{3})^n$ and $\sum_{n =0}^{ \infty} (\dfrac{1}{3})^n$ Both series are a convergent geometric series with first term, $a=1$ and common ratio $r =\dfrac{2}{3}, \dfrac{1}{3}$ Thus, $S=\dfrac{1}{1-\dfrac{2}{3}}-\dfrac{1}{1-\dfrac{1}{3}}$ or, $=\dfrac{3}{3-1}$ or, $=\dfrac{3}{2}$