The sum to infinity of the series $1 + \frac{1}{2} + \frac{1}{2 \cdot 4} + \frac{1}{2 \cdot 4 \cdot 8} + ...$is
Step-by-Step Solution
Key Concept: Using the method of multiplying by a constant and subtracting to convert a series with linearly increasing numerators into a geometric series.
Given $S = 1 + \frac{4}{3} + \frac{7}{9} + \frac{10}{27} + \cdots$ and $\frac{S}{3} = 0 + 1 + \frac{4}{3} + \frac{7}{9} + \cdots$. Subtracting equation (2) from equation (1): $\frac{2S}{3} = 1 + \frac{1}{3} + \frac{1}{9} + \cdots = \frac{1}{1-1/3} = \frac{3}{2}$. Therefore, $S = \frac{3}{2} \times \frac{3}{2} = \frac{48}{16}$.
Correct Answer: 48/16