Maclaurin Series

A Maclaurin series is a specific type of Taylor series that is centered at $a=0$. It provides a way to represent a function as an infinite sum of terms involving powers of $x$.

The Formula

If a function $f(x)$ is infinitely differentiable at $0$, its Maclaurin series is:

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

Why Use Maclaurin Series?

Since many fundamental functions (like $e^x$, $\sin (x)$, and $\cos (x)$) are well-behaved at $x=0$, the Maclaurin series is often the simplest and most common way to approximate these functions near the origin.

Common Maclaurin Series

Function	Series Expansion
$e^x$	$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$
$\sin (x)$	$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots$
$\cos (x)$	$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots$
$\frac{1}{1-x}$	$1+x+x^2+x^3+\dots$ (for $

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Draft note: Connect this to the Taylor Series post for a complete overview.