1962 AHSME Problem 40

The limiting sum of the infinite series, \frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \ldots whose nth term is \frac{n}{10^n} is:

\textbf{(A)}\ \frac{1}9\qquad \textbf{(B)}\ \frac{10}{81}\qquad \textbf{(C)}\ \frac{1}8\qquad \textbf{(D)}\ \frac{17}{72}\qquad \textbf{(E)}\ \infty

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.