$\frac{1}{3} + \frac{1}{6} + \frac{1}{12} + \frac{1}{24} + \frac{1}{48}$
$= \frac{1}{3} + \frac{1}{6} + \frac{1}{12} + \frac{1}{24} + \frac{1}{48} + \frac{1}{48} - \frac{1}{48}$
$= \frac{1}{3} + \frac{1}{6} + \frac{1}{12} + \frac{1}{24} + \frac{1}{24} - \frac{1}{48}$
$= \frac{1}{3} + \frac{1}{6} + \frac{1}{12} + \frac{1}{12} - \frac{1}{48}$
$= \frac{1}{3} + \frac{1}{6} + \frac{1}{6} - \frac{1}{48}$
$= \frac{1}{3} + \frac{1}{3} - \frac{1}{48}$
$= \frac{2}{3} - \frac{1}{48}$
$= \frac{31}{48}$