This preview shows page 1. Sign up to view the full content.
Unformatted text preview: e convergence at endpoints But then Example 1: ﬁnd the radius of
convergence of the power series
∞
n=0 4n
an
= n+1
an+1
4 n (−4)
√
xn .
n+3 (n + 1) + 3
1
√
=
4
n+3 n+4
.
n+3 Consequently,
∞ an xn Solution: this is a power series
with n=0 R = lim n→∞ an
1
= lim
n→∞ 4
an+1 n+4
1
=.
n+3
4 (−4)n
4n
an = √
= (−1)n √
n+3
n+3 ∞ an xn be a general To begin to understand what the Radius of convergence does for us, let
power series. Whenever the series converges it deﬁnes a function n=0 ∞ an xn . f (x) = a0 + a1 x + a2 x2 + a3 x3 + . . . + an xn + . . . =
n=0 It’s certainly deﬁned at x = 0, for instance, since f (0) = a0 . Deﬁnition: the Interval of Convergence of a power series n an xn is the largest interval on which the series is deﬁned; it is the domain of
th...
View
Full
Document
This note was uploaded on 06/05/2011 for the course MATH 408 D taught by Professor Gilbert during the Spring '11 term at University of Texas at Austin.
 Spring '11
 Gilbert
 Power Series

Click to edit the document details