AHSME 1952 Problem 50

A line initially 1 inch long grows according to the following law, where the first term is the initial length. $$1+\frac{1}{4}\sqrt{2}+\frac{1}{4}+\frac{1}{16}\sqrt{2}+\frac{1}{16}+\frac{1}{64}\sqrt{2}+\frac{1}{64}+\cdots$$ If the growth process continues forever, the limit of the length of the line is: $\textbf{(A) } \infty \textbf{(B) } \frac{4}{3} \textbf{(C) } \frac{8}{3} \textbf{(D) } \frac{1}{3}(4+\sqrt{2}) \textbf{(E) } \frac{2}{3}(4+\sqrt{2})$