I need to find $ $ S = \sum_{n=1}^{\infty}{\frac{1}{n 2^{n-1}}}$ $

**Attempt:**

$ $ f'(x) = \sum_{n=1}^{\infty}\frac{x^n}{2^n} = \frac{x}{2-x}$ $

Which is just evaluating geometric series

$ $ f(x) = \sum_{n=1}^{\infty}\frac{x^{n+1}}{(n+1)2^{n}}$ $

Now, by finding antiderivative of $ \frac{x}{2-x}$

$ $ f(x) = -x-2\ln(x-2)$ $

Finding sum should be just $ $ S = 1 + f(1)$ $

But $ f(x)$ is undefined as a real-valued function for $ x \leq 2$