\[ f(x) = \frac{1}{(x-1)} about\ x = 0 \\f(x) = \sum_{n=0}^{\infty}x^n \]
\[ f(x) = e^x \\ c_{n} = \frac{f^{(n)}(0)}{n!} = \frac{1}{n!} \\ e^x = \sum_{n=0}^{\infty}\frac{1}{n!}x^n \]
\[ f(x) = ln(1+x) at\ a = 0 \\ = ln(1+0) + \frac{\frac{d}{dx}(ln(1+x))(0)}{1!} + \frac{\frac{d^2}{dx^2}(ln(1+x))(0)}{2!} + \frac{\frac{d^3}{dx^3}(ln(1+x))(0)}{3!} + ... \\ = \sum_{n=0}^{\infty}(-1)^{n+1}\frac{x^n}{n} \]