Taylor's Formula

We can always find a power series expansion (centered at $x=a$) of an infinitely differentiable function using the following rule:

$$f(x)=\displaystyle \sum^{\infty}_{n=0}\displaystyle \frac{f^{(n)}(a)(x-a)^n}{n!}$$

If the series is centered about $x=0$ we get a McLaurin series

$$f(x)=\displaystyle \sum^{\infty}_{n=0}\displaystyle \frac{f^{(n)}(0)x^n}{n!}$$

We can use this to find the following series:

$$\sin(x) = \displaystyle \sum^{\infty}_{n=0}(-1)^n\displaystyle \frac{x^{2n+1}}{(2n+1)!}$$

$$e^x = \displaystyle \sum^{\infty}_{n=0}\displaystyle \frac{x^n}{n!}$$

$$\ln(1+x) = \displaystyle \sum^{\infty}_{n=0}(-1)^n\displaystyle \frac{x^n}{n}$$