Calculus Notes 12.8

Calculus Notes 12.8 - Calculus-Stewart Dr Berg Summer 09...

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Calculus- Stewart Dr. Berg Summer ‘09 Page 1 12.8 12.8 Power Series Definition Any series of the form a k x k k = 0 or a k x a ( ) k k = 0 is called a power series in x (centered at a ). Note: a k x a ( ) k k = 0 is just a translation of a k x k k = 0 , so we mostly work with a k x k k = 0 . Lemma Let c and d be real numbers such that c < d . If a k d k k = 0 converges then a k c k k = 0 converges absolutely, and if a k c k k = 0 diverges then a k d k k = 0 diverges. Proof: If a k d k k = 0 converges then a k d k 0 , so for some K N , a k d k 1 for all k K . Thus a k c k = a k d k k 1 k = k . Also, if c < d we have < 1 . Hence, 0 a k c k k is a convergent geometric series, so a k c k k = 0 converges absolutely. An easy consequence of this is the following. Theorem For a given power series a k x a ( ) k k = 0 there are three possible cases. i) The series converges only when x = a . ii) The series converge for all real numbers x . iii) There is a positive number R such that the series converges absolutely for x a < R and diverges for x
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This note was uploaded on 06/06/2010 for the course M 408 taught by Professor Hodges during the Spring '08 term at University of Texas.

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Calculus Notes 12.8 - Calculus-Stewart Dr Berg Summer 09...

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