Taylor series

Definition

A Taylor series is an infinite sum of terms that represents a function as a series of its derivatives evaluated at a single point. The series converges to the function within a certain interval around that point.

5 Must Know Facts For Your Next Test

1. The general form of a Taylor series for a function $f(x)$ centered at $a$ is given by $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$
2. A Maclaurin series is a special case of the Taylor series centered at $a = 0$.
3. The radius of convergence determines the interval within which the Taylor series converges to the function.
4. Taylor polynomials are finite sums that approximate functions and are derived from truncating the Taylor series.
5. If all terms beyond a certain degree in the Taylor series are zero, then the original function is a polynomial.

Review Questions

• What is the general form of a Taylor series?
• What distinguishes a Maclaurin series from other Taylor series?
• How do you determine the radius of convergence for a Taylor series?

Related terms

Maclaurin Series:

A special case of the Taylor series centered at $a = 0$, represented as $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$

Radius of Convergence:

The distance from the center point $a$ within which the Taylor or Maclaurin series converges to its corresponding function.

Taylor Polynomial:

A finite sum derived from truncating the Taylor series, used to approximate functions over an interval.