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

• What is the general form of a Maclaurin series?
• How do you derive the Maclaurin series for a given function?
• What is the radius of convergence in relation to a Maclaurin series?

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.