Discussion
The sum of the following infinite series is
\[ \begin{aligned} &2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{8} \\ &+ \frac{1}{9} + \frac{1}{16} + \frac{1}{27} + \cdots \end{aligned} \]
Answer: B
The sum is 7/2.
Apart from the leading 2, the terms are the powers of 1/2 and the powers of 1/3 interleaved. Summing each geometric series separately:
\[ \begin{aligned} \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots &= \frac{1/2}{1 - 1/2} = 1 \\ \tfrac{1}{3} + \tfrac{1}{9} + \tfrac{1}{27} + \cdots &= \frac{1/3}{1 - 1/3} = \tfrac{1}{2} \\ 2 + 1 + \tfrac{1}{2} &= \tfrac{7}{2} \end{aligned} \]
Rearranging is allowed because every term is positive. 11/3, 13/4 and 9/2 come from mis-summing one of the two series.
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