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Theorem 1: Convergence and Divergence

THEOREM 1

(i) If a power series

09_infinite_series-431.gif

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 09_infinite_series-432.gif 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 09_infinite_series-433.gif converges,

(2)09_infinite_series-434.gif

Now by the Limit Comparison Test, 09_infinite_series-435.gif converges absolutely,

(ii) This follows trivially from (i).

Let 09_infinite_series-436.gif diverge and |u| > |v|.

09_infinite_series-437.gif cannot converge because if it did 09_infinite_series-438.gif would convergeabsolutely. Therefore 09_infinite_series-439.gif 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 09_infinite_series-440.gif, 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.

09_infinite_series-441.gif

Figure 9.7.1
Home Infinite Series Power series Theorem 1: Convergence and Divergence

Last Update: 2006-11-08