Compute the infinite sum $S,$ where
\[S = \frac{1}{2} - \frac{2}{4} + \frac{3}{8} - \frac{4}{16} + \frac{5}{32} - \dots + \frac{n}{2^n} (-1)^{n + 1} + \dotsb.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute the infinite sum $S,$ where
\[S = \frac{1}{2} - \frac{2}{4} + \frac{3}{8} - \frac{4}{16} + \frac{5}{32} - \dots + \frac{n}{2^n} (-1)^{n + 1} + \dotsb.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{2}{9}
Compute the infinite sum $S,$ where
\[S = \frac{1}{2} - \frac{2}{4} + \frac{3}{8} - \frac{4}{16} + \frac{5}{32} - \dots + \frac{n}{2^n} (-1)^{n + 1} + \dotsb.\]
train