## Theorem 1: Convergence and Divergence

THEOREM 1

(i) If a power series

converges when x = u, then it converges absolutely whenever |x| < |u|.

(ii) If a power series diverges when x = v, then it diverges whenever |x| > |v|.

PROOF

(i) Suppose the series converges.

Then for any positive infinite H, aHuH is infinitesimal. Let |v| < |u|. The ratio b = |u|/|u| is then less than one. It follows that:

(1) The positive term geometric series converges,

(2)

Now by the Limit Comparison Test, converges absolutely,

(ii) This follows trivially from (i).

Let diverge and |u| > |v|.

cannot converge because if it did would convergeabsolutely. Therefore diverges.

Theorem 1 shows that if a power series converges at x = u and at x = v, then it converges absolutely at every point strictly between u and v. We conclude that the set of points where the power series converges is an interval, called the interval of convergence. (A rigorous proof that the set is an interval is given in the Epilogue.) The next corollary summarizes what we know about the 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

Last Update: 2006-11-08