\begin{gather*} f'(x) = \sum_{k=1}^{\infty} {kx^{k-1}} = \frac{1}{(1-x)^2}\\ f''(x) = \sum_{k=2}^{\infty} {k(k-1)x^{k-1}} = \frac{2}{(1-x)^3}\\ f^{(n)}(x) = \sum_{k=n}^{\infty} {k(k-1)...(k-n+1)x^{k-n}} = \frac{n!}{(1-x)^{n+1}}\\ \int f(x) dx = \sum_{k=0}^{\infty} {\frac{x^{k+1}}{k+1}} = -ln(1-x) \end{gather*}