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# PP 12.10 - Taylor and Maclaurin Series Objectives To...

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Taylor and Maclaurin Series

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Objectives To determine which functions have power series representations. To establish a method for finding such power series representations. To determine the radius of convergence of the power series representations of commonly used functions.
Finding a Power Series Representation n n n a) (x c f(x) - = = 0 Assume ... ) ( ) ( ) ( 3 3 2 2 1 0 + - + - + - + = a x c a x c a x c c f(x) 0 c f(a) = ... ) ( 3 ) ( 2 ) ( 2 3 2 1 + - + - + = a x c a x c c x f 1 ) ( c a f =

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... ) ( ) 3 ( 4 ) ( ) 2 ( 3 2 ) ( 2 4 3 2 + - + - + = a x c a x c c x f 2 ) ( 2 ) ( 2 2 a f c c a f = = ... ) )( 3 )( 4 ( 5 ) ( ) 2 )( 3 ( 4 ) 2 ( 3 ) ( 2 4 3 + - + - + = a x a x c c x f ... ) ( 4 ) ( 3 ) ( 2 ) ( 3 4 2 3 2 1 + - + - + - + = a x c a x c a x c c x f ) 2 ( 3 ) ( ) 2 ( 3 ) ( 3 3 a f c c a f = =
Proceeding as above, we get that ! ) ( ) 1 )( 2 ( 3 ) 2 )( 1 ( ) ( ) ( ) ( n a f n n n a f c n n n = - - = ! ) ( then , if Hence, ) ( 0 n a f c a) (x c f(x) n n n n n = - = =

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Taylor Series of a Function n n n a) (x n a f f(x) - = = 0 ) ( ! ) ( ). ( of series Maclaurin the is ! ) 0 ( , 0 When 0 ) ( x f x n f f(x) a n n n = = =
Comments The Taylor series is named after the English mathematician Brook Taylor (1685–1731).

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