, 2013

Solution:
The decimal representation of $\frac{1}{n}$ terminates if and only if $n = 2^{i} 5^{j}$ for some nonnegative integers $i, j$, so our desired sum is
$$\sum_{i \geq 0} \sum_{j \geq 0} 2^{-i} 5^{-j} = \sum_{i \geq 0} 2^{-i} \sum_{j \geq 0} 5^{-j} = \left(1 - 2^{-1}\right)^{-1} \left(1 - 5^{-1}\right)^{-1} = \frac{2}{1} \frac{5}{4} = \frac{5}{2}$$