Answer: B

Small Hint:

Group the terms in consecutive blocks of three

Big Hint:

Each block is $\frac{1}{8}$ times the preceding block

Solution:

Group the series as $$\begin{aligned} S={}&\left(1-\frac12-\frac14\right)\\ &+\left(\frac18-\frac1{16}-\frac1{32}\right) +\cdots. \end{aligned}$$ The first block is $\frac{1}{4},$ and successive blocks form a geometric series with ratio $\frac{1}{8}.$ Hence $$S=\frac{\frac{1}{4}}{1-\frac{1}{8}}=\frac27.$$

Therefore, the correct answer is B.