Taylor Series

A Taylor series is a way to represent a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point.

The Taylor series of a function $f(x)$ about a point $a$ is given by the formula:

$$f(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\frac{f^{\prime \prime \prime }(a)}{3!}(x-a{)}^3+\dots =\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

Example $e^x$

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