Notes on the Radius of Convergence for a Power Series

Notes on the Radius of Convergence for a Power Series -...

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Notes on the Radius of Convergence for a Power Series Definition : A power series is a series of the form ( 29 0 n n n c x a = - , where a is a constant, { } 0 n n c = is a sequence of numbers and x is a variable. Sometimes a power series is referred to as a power series centered about a (or a power series centered at a ). Examples : Below are several examples of power series in which the various components of the series are identified. It is extremely important that are able to identify the components of a power series. 1) Consider the power series 0 n n x = : For this series 0 a = and for all 0 n , 1 n c = . 2) Consider the power series ( 29 0 1 ! n n x n = - : For this series 1 a = and for all 0 n , 1 ! n c n = . Remember ! 1 2 3 n n = ⋅ ⋅ L . 3) Consider the power series ( 29 0 ! 3 n n n x = + : For this series 3 a = - , and for all 0 n , ! n c n = . 4) Consider the power series ( 29 ( 29 1 2 1 n n n x n = - - : For this series 2 a = , 0 0 c = and for 1 n , ( 29 1 n n c n - = . 5) Consider the power series ( 29 2 2 1 n n n x = + : For this series 1 a = - , 0 0 c = , 1 0 c = and for all 2 n , 2 n c n = . 6) Consider the power series ( 29 2 1 2 1 n n x n = - : This series does not appear to be a power series at all since it does not have the proper form defined above. However with the help of algebra,
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( 29 2 2 2 2 1 1 2 2 2 1 2 2 1 2 2 n n n n n n x x x x n n n n - - - = = = - .So we may write the series as ( 29 2 2 1 1 2 1 2 1 2 n n n n n x x n n = = - = - . Then 1 2 a = , 0 0 c = and for all 1 n , 2
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This note was uploaded on 05/06/2011 for the course MATH 256 taught by Professor Buekman during the Spring '11 term at Purdue.

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Notes on the Radius of Convergence for a Power Series -...

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