## Corollary: Interval of Convergence

COROLLARY

For each power series , one of the following happens.

(i) The series converges absolutely at x = 0 and diverges everywhere else.

(ii) The series converges absolutely on the whole real line (-∞, ∞).

(iii) The series converges absolutely at every point in an open interval (-r, r) and diverges at every point outside the closed interval [-r, r]. At the endpoints -r and r the series may converge or diverge, so the interval of convergence is one of the sets

(-r,r), [-r, r), (-r, r], [-r, r].

Figure 9.7.1 illustrates part (iii) of the Corollary. The number r is called the radius of convergence of the power series. In case (i) the radius of convergence is zero, and in case (ii) it is ∞. Once the radius of convergence is determined, we need only test the series at x = r and x = -r to find the interval of convergence.

Figure 9.7.1

 Example 1: Interval Of Convergence
 Example 2: Interval of Convergence (Half-open)
 Example 3: Interval of Convergence (-∞, ∞)

If we replace x by x - c we obtain a power series in x - c,

The power series has the same radius of convergence as

and the interval of convergence is simply moved over so that its center is c instead of 0.

For example, if has interval of convergence (- r, r], then

has interval of convergence (c - r, c + r], illustrated in Figure 9.7.2.

Figure 9.7.2

 Example 5: Interval of Convergence

Last Update: 2006-11-08