9. Problem: Evaluate $\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^{n-1}}$.
Solution: Note that if we take the integral of $f(x)$ in problem 4 , we get the function $F(x)=x+\frac{x^{2}}{2 \cdot 2}+$ $\frac{x^{3}}{3 \cdot 2^{2}}+\ldots$. Evaluating this integral in the interval $[0,1]$, we get $1+\frac{1}{2 \cdot 2}+\frac{1}{3 \cdot 2^{2}}+\ldots$, which is the desired sum. Hence $\int_{0}^{1} \frac{2}{2-x} d x=2 \ln 2$.