---
tag: 043SJ5RF59
kind: definition
name: Maclaurin series
msc: [41A58]
defines: 0K7P6PR838
uses: [0225QPWRRE]
sources:
  - url: https://en.wikipedia.org/wiki/Taylor_series
    locator: Definition
---

## Definition

Let $I$ be an open interval with $0 \in I$, and let $f \colon I \to \mathbb{R}$ be smooth.
The Maclaurin series of $f$ is its [[Taylor series|0225QPWRRE]] at $0$:
$$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n.$$

## Remarks

For example, the Maclaurin series of $e^x$ is $\sum x^n / n!$, and the one of $1/(1 - x)$ is $\sum x^n$.
The series is named after Colin Maclaurin.

## Explanations

### L1 human

A Maclaurin series is a Taylor series centered at $0$.