jmc

Multiplying the sum by $\frac{1}{2},$ we get $$\frac{1}{2}S=\frac{1}{4}-\frac{2}{8}+\frac{3}{16}-\frac{4}{32}+\cdots .$$Then $$\begin{array}{rl}S+\frac{1}{2}S& =\left(\frac{1}{2}-\frac{2}{4}+\frac{3}{8}-\frac{4}{16}+\frac{5}{32}-\cdots \right)+\left(\frac{1}{4}-\frac{2}{8}+\frac{3}{16}-\frac{4}{32}+\cdots \right)\\ & =\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\cdots \\ & =\frac{1/2}{1+1/2}=\frac{1}{3}.\end{array}$$This gives us $\frac{3}{2}S=\frac{1}{3},$ so $S=\boxed{\frac{2}{9}}.$