# In the case of the power series for ex 1n x n n0 in

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: his limit exists. Let us check that the ratio test gives the right answer for the radius of convergence of the power series (1.2). In this case, we have an = 1 , bn so |an | bn+1 1/bn = n = b, = |an+1 | 1/bn+1 b and the formula from the ratio test tells us that the radius of convergence is R = b, in agreement with our earlier determination. In the case of the power series for ex , ∞ 1n x, n! n=0 in which an = 1/n!, we have |an | 1/n! (n + 1)! = = = n + 1, |an+1 | 1/(n + 1)! n! and hence R = lim n→∞ |an | = lim (n + 1) = ∞, |an+1 | n→∞ so the radius of convergence is inﬁnity. In this case the power series converges for all x. In fact, we could use the power series expansion for ex to calculate ex for any choice of x. On the other hand, in the case of the power series ∞ n!xn , n=0 4 in which an = n!, we have |an | n! 1 = = , |an+1 | (n + 1)! n+1 |an | = lim |an+1 | n→∞ R = lim n→∞ 1 n+1 = 0. In this case, the radius of convergence is zero, and the power series does not conver...
View Full Document

## This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online