chap11sec9

chap11sec9 - Example: Find the power series representation...

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11.9 Representations of Functions as Power Series To represent a function as a sum of a power series we manipulate a geometric series or we differentiate or integrate it. Remember from our study of geometric series that  0 | | 1 1 n n a ar r r = = < - Similarly, this tells us from the perspective of a power series that 0 1 , 1 1 1 1 n n x r x a when x x = = = = - < < - . So, the function  1 ( ) 1 f x x = -  can be represented as a power series for part of its domain.  Look at the graphs below.  The graph of the series is on the left and the function graph is on the right. Notice that the graphs   are the same. 100 0 1 1 n n x for x = - < < 1 ( ) 1 f x x = -      -1 -0.5 0.5 1 1 2 3 4 5                         -1 -0.5 0.5 1 1 2 3 4 5 In similar ways other functions can be represented by power series.  
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Unformatted text preview: Example: Find the power series representation of 4 1 ( ) 16 f x x = + . Differentiation and integration are useful techniques for finding power series representations of functions. Differentiation and integration of power series works in a way very similar to handling polynomials. Look at the theorem on page 746 and the notes on page 747. Using this theorem we will be able to find a power series representation of functions. Example: Use differentiation to find the power series representation for 2 1 ( ) ( 3) f x x =-. Example: Use integration to find the power series representation of ( ) ln( 1) f x x = + ....
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This note was uploaded on 01/20/2012 for the course MATH 2057 taught by Professor Estrada during the Fall '08 term at LSU.

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