Taylor Series

A function about a point $a$ can be expanded as:

$$f(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\frac{f^{\prime \prime \prime }(a)}{3!}(x-a{)}^3+\cdots +\frac{f^{(n)}(a)}{n!}(x-a{)}^n+\cdots$$

or in summation form

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

This lets us approximate functions near some operating point $a$ (the farther we are from $a$, the less accurate the approximation).