AHSME 1962 Problem 40

The limiting sum of the infinite series, $\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots$ whose $n$th term is $\frac{n}{10^n}$ is: $\textbf{(A)}\ \frac{1}9 \textbf{(B)}\ \frac{10}{81} \textbf{(C)}\ \frac{1}8 \textbf{(D)}\ \frac{17}{72} \textbf{(E)}\ \text{larger than any finite quantity}$