\[f'(x) = \frac{1}{(1-x)^2} \] \[f''(x) = \frac{-2}{(1-x)^3} \] \[f'''(x) = \frac{6}{(1-x)^4} \]
\[\frac{1}{(1-x)} = 1 +x +x^2 +x^3 +...\] \[\frac{1}{(1-x)} = \frac{1}{(1-x)}\sum_{n=0}^{\infty}x^n \]
\[f'(x)=e^x\] \[f''(x)=e^x\]
\[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +...\] \[e^x = \sum_{n=1}^{\infty} \frac{x^n}{n!} \]
\[f'(x)=\frac{1}{(x+1)}\]
\[f''(x)=\frac{-1}{(x+1)^2}\]
\[f'''(x)=\frac{2}{(x+1)^3}\]
\[ln(1+x) = x - \frac{1}{2}x^2 + \frac{2}{3}x^3 + ...\]
\[ln(1+x) = \sum_{n=1}^{\infty} -1^{n+1} \frac{x^n}{n}\]