This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Power Series
John E. Gilbert, Heather Van Ligten, and Benni Goetz What’s the payoﬀ for introducing all these various tests and applying them to complicated series?
Well, think of the remarkable series
x2 x3 x4
xn
xn
= 1+x+
+
+
+ ... +
+ . . . = ex
n!
2!
3!
4!
n! ∞
n=0 we mentioned right at the beginning. Of course, we still have no idea why the sum of the series is ex ;
that’s coming up soon! But at least now we see that the series converges absolutely for all x because
by the Ratio test
lim n→∞ an+1
an = lim n→∞ xn+1
xn x
n!
= lim
=0
n→∞ n
(n + 1)! for each x. The series thus has a ﬁnite sum for each x, even if we haven’t actually determined the sum
of the series or know why absolute convergence helps.
Now let’s look at Geometric series and Har...
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