jmc

Let the sum be $S$. We can split this into 2 geometric series: $$\begin{array}{rl}S& =\left(\frac{1}{7}+\frac{1}{7^2}+\frac{1}{7^3}+\cdots \right)+\left(\frac{1}{7^2}+\frac{1}{7^4}+\frac{1}{7^6}+\cdots \right)\\ & =\frac{1/7}{1-1/7}+\frac{1/49}{1-1/49}\\ & =\frac{1/7}{6/7}+\frac{1/49}{48/49}\\ & =\frac{1}{6}+\frac{1}{48}\\ & =\boxed{\frac{3}{16}}\end{array}$$