We already know the power series for $\displaystyle\frac{1}{1+x}$, $\ln(1+x)$, $\displaystyle\frac{1}{(1-x)^2}$, and $\tan^{-1}(x)$. Using the representation formula in Taylor's Theorem,
$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n,$$ we can derive the Taylor series expansion for a number of other common functions: