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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 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...
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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.

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