Problem 44 - Essential Calculus

Question

Describe the radius of convergence of a power series. Describe the interval of convergence of a power series.

Step-by-Step Solution

Answer

The radius of convergence R for a power series is a number that determines the interval around the center c where the series converges. The interval of convergence is the set of all x for which the series converges, including possible endpoints \(x = c - R\) and \(x = c + R\), determined by evaluating the series at those points.

Step 1: Defining series and radius of convergence

A power series is a series of the form \(\sum_{i=0}^{\infty} a_i (x-c)^i\), where the \(a_i\) are constants and c is the center of the series. The radius of convergence R is a non-negative number or infinity such that the series converges if \(0 \leq |x-c| < R\) and diverges if \(|x-c| > R\). The exact behavior at \(|x-c| = R\) is not determined by the radius of convergence alone and must be verified separately.

Step 2: Defining interval of convergence

The interval of convergence is the set of all x for which the series converges. This is a subset of the interval \((c-R, c+R)\), and may include one or both of the endpoints \(c-R, c+R\). To determine if the endpoints are in the interval of convergence, we can substitute \(x = c-R\) and \(x = c+R\) into the power series and check if the resulting series converge.

Step 3: Checking endpoints

For each of the endpoints, plug in the value \(x = c - R\) or \(x = c + R\) into the series. This gives a new series, which might be easier to work with. If these series converge, then the endpoints are included in the interval of convergence. If they diverge, the endpoints are not included.