Maclaurin series

Definition

A Maclaurin series is a special case of the Taylor series, centered at zero. It represents a function as an infinite sum of its derivatives at zero.

5 Must Know Facts For Your Next Test

1. The general form of a Maclaurin series is $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$.
2. Maclaurin series are useful for approximating functions near $x=0$.
3. Common functions with known Maclaurin series include $e^x$, $\sin(x)$, and $\cos(x)$.
4. The radius of convergence for a Maclaurin series determines where the series accurately represents the function.
5. Maclaurin series can be derived by differentiating the function multiple times and evaluating these derivatives at zero.

Review Questions

Related terms

Taylor Series:

A representation of a function as an infinite sum of terms calculated from its derivatives at a single point.

Radius of Convergence:

The distance from the center point within which a power series converges.

Power Series:

$\sum_{n=0}^{\infty} c_n (x-a)^n$, where $c_n$ are coefficients and $(x-a)$ is raised to increasing powers.