Maclaurin Series

Summary

A Maclaurin series is a Taylor series centered at $0$ . Many elementary functions have simple Maclaurin expansions used for approximation and analysis.

Prerequisites

Taylor Series, Power Series, Higher-Order Derivatives

Formula

If $f$ is infinitely differentiable at $0$ , its Maclaurin series is

f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n},

when the series equals $f$ on an interval (checked via remainder theorems).

Worked Example

e^{x} = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}, \sin x = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n + 1}}{(2 n + 1)!}, \cos x = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n}}{(2 n)!} .

Each of these converges for all real $x$ (radius $R = \infty$ ).

Common Mistakes

Assuming every $C^{\infty}$ function equals its Maclaurin series (counterexamples exist; check remainders).
Using degree-truncated polynomials without an error bound when accuracy matters.

Connections

General center: Taylor Series
Special case: Binomial Series for $(1 + x)^{α}$

References

Maclaurin series are in OpenStax Calculus Volume 2.^[1]

OpenStax, Calculus Volume 2, Section 6.3, https://openstax.org/details/books/calculus-volume-2 ↩︎