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Home Infinite Series Derivatives and Integrals of Power Series Theorem 2: Radius of Convergence
 
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Theorem 2: Radius of Convergence

We are now ready to prove the power series formulas for 1/(1 - x) and ex.

THEOREM 2

(i) 09_infinite_series-517.gif, r = 1.

(ii) 09_infinite_series-518.gif, r = ∞.

PROOF

(i) is just the geometric series for x. We proved in Section 9.2 that it converges to 1/(1 - x) for |x| < 1 and diverges for |x| ≥ 1.

(ii) Let

09_infinite_series-519.gif

At x = 0 we have y = 1. We can find dy/dx by Theorem 1.

09_infinite_series-520.gif

The radius of convergence is ∞, so for all x,

09_infinite_series-521.gif

The general solution of this differential equation (see Section 8.6) is

y = Cex. At x = 0, 1 = Ce0 = C. Therefore y = ex.
Home Infinite Series Derivatives and Integrals of Power Series Theorem 2: Radius of Convergence

Last Update: 2006-11-08