Maclaurin Series

A way of representing a function as a Power Series. It is a Taylor Series at $a=0$ for $f$.

Formal Definition

$f(x)={\sum }_{n=1}^{\infty }\frac{f^n(0)}{n!}(x-0{)}^n$ is a maclaurin series representation of $f$

Example

Compute the MS for $f(x)=e^x$

Soln

1. ${\sum }_{n=1}^{\infty }{f}^n(0)x^n$ by defn of MS
2. We know that $f^n(x)=e^x$ so, $f^n(0)=1$ by defn of $e^x$
3. Thus, ${\sum }_{n=1}^{\infty }\frac{x^n}{n!}$ is our MS for $f(x)=e^x$