smc

Notice that we can arrange the sequence like so: $$\begin{array}{rl}1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots & =1-\frac{5}{8}-\frac{5}{64}-\frac{5}{512}-\dots \\ & =1-\left(\sum _{i=1}^{\infty }\frac{5}{8^n}\right)\\ & =1-\frac{5}{7}\text{ (by the convergence of a geometric series)}\\ & =\frac{2}{7}\end{array}$$ hence our answer is $\boxed{\frac{2}{7}}$