key term - Interval of convergence

Definition

The interval of convergence is the set of all real numbers for which a given power series converges. It includes the radius of convergence and specifies whether the endpoints are included or excluded.

5 Must Know Facts For Your Next Test

1. The interval of convergence can be found using the ratio test or root test.
2. The radius of convergence is the distance from the center point to either endpoint of the interval.
3. Endpoints must be tested separately to determine if they are part of the interval of convergence.
4. A power series converges absolutely within its radius of convergence.
5. The interval can be finite, infinite, or a single point depending on the series.

Review Questions

Related terms

Power Series:

An infinite series in the form $\sum_{n=0}^{\infty} a_n (x - c)^n$, where $a_n$ represents coefficients and $c$ is the center.

Radius of Convergence:

The distance from the center $c$ to any boundary point within which a power series converges absolutely.

Ratio Test:

A method used to determine whether a series converges by examining the limit $\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right|$. If this limit is less than 1, the series converges absolutely.