Taylor and Maclaurin Series

The Formula for Taylor Series

A function is called analytic if it can be expressed as an infinite power series around some point $a$. Taylor's theorem gives us a formula for the coefficients of the power series expansion of an analytic function:

Taylor's Theorem

If $\displaystyle f(x) = \sum_{n=0}^\infty c_n (x-a)^n$, then $\displaystyle c_n = \frac{f^{(n)}(a)}{n!}$.

This formula works both ways: if we know the $n$-th derivative, we can figure out $c_n$; if we know $c_n$, we can figure out the $n$-th derivative.