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Unformatted text preview: Power Series
John E. Gilbert, Heather Van Ligten, and Benni Goetz What’s the pay-oﬀ for introducing all these various tests and applying them to complicated series?
Well, think of the remarkable series
x2 x3 x4
+ ... +
+ . . . = ex
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
(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...
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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
- Power Series