The sum of the following infinite series is
\[2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} + \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \cdots\]
Answer
(B)
Explanation
After the leading 2, the terms are two geometric series interleaved: the powers of \(\tfrac{1}{2}\) (\(\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \tfrac{1}{16}, \ldots\)) and the powers of \(\tfrac{1}{3}\) (\(\tfrac{1}{3}, \tfrac{1}{9}, \tfrac{1}{27}, \ldots\)). All terms are positive, so they may be regrouped. With \(\frac{a}{1 - r}\) for each series:
\[\frac{1/2}{1 - 1/2} = 1, \qquad \frac{1/3}{1 - 1/3} = \frac{1}{2}.\]
Sum \(= 2 + 1 + \tfrac{1}{2} = \tfrac{7}{2}\).
Answer: B