8-Power Series

8-Power Series - Power Series John E Gilbert Heather Van...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ∞ n = 0 x n n ! = 1 + x + x 2 2! + x 3 3! + x 4 4! + . . . + x n n ! + . . . = e x we mentioned right at the beginning. Of course, we still have no idea why the sum of the series is e x ; 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 →∞ a n +1 a n = lim n →∞ x n +1 x n n ! ( n + 1)! = lim n →∞ x n = 0 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 Harmonic series in this context. Thinking of the common ratio as a variable x , not as a number r , we get f ( x ) = ∞ n = 0 x n = 1 + x + x 2 + x 3 + x 4 + . . . + x n + . . . = 1 1- x (take a = 1 for simplicity). The series converges absolutely on (- 1 , 1) and on this interval its sum is 1 1- x , so the series provides a series representation of a simple function from high school days. But this is calculus, so why not integrate both the Geometric series and its sum: x ∞ n = 0 t n dt = ∞ n = 0 x t n dt = ∞ n = 0 x n +1 n + 1 = ∞ n = 0 x n n x 1 1- t dt =- ln(1- x ) = ln 1 1- x , assuming that the integral of the infinite sum is the sum of the infinite number of integrals ; that’s coming up too! Thus ln 1 1- x = x + x 2 2 + x 3 3 + x 4 4 + . . . + x n n + . . . , which on letting x → 1- becomes ln( ∞ ) = 1 + 1 2 + 1 3 + 1 4 + . . . + 1 n + . . . = ∞ ....
View Full Document

This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.

Page1 / 6

8-Power Series - Power Series John E Gilbert Heather Van...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online