Radius of convergence

Definition

The radius of convergence is the distance within which a power series converges to a finite value. It determines the interval around the center point where the series is valid.

5 Must Know Facts For Your Next Test

1. The radius of convergence can be found using the ratio test or root test.
2. If $R$ is the radius of convergence, then the power series converges for all $x$ such that $|x - a| < R$, where $a$ is the center of the series.
3. Outside the interval $(a - R, a + R)$, the power series diverges.
4. At the endpoints of this interval, convergence must be checked separately as it could either converge or diverge.
5. The radius of convergence can be zero, indicating that the series only converges at its center.

Review Questions

• How do you determine if a given power series converges within a certain interval?
• What tests can be used to find the radius of convergence?
• What must be checked separately when evaluating convergence at endpoints?

Related terms

Power Series:

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

Ratio Test:

A method to determine convergence or divergence of an infinite series by examining $\lim_{n \to \infty} |a_{n+1}/a_n|$.

Root Test:

A method to determine convergence by examining $\lim_{n \to \infty} \sqrt[n]{|a_n|}$.