MTH 253 Calculus (Other Topics)Taylor & Maclaurin PolynomialsTaylor & Maclaurin SeriesPower SeriesConvergence Power SeriesSlide 6Power Series: ExampleSlide 8Power Series in x–x0Convergence Power Series in x-x0MTH 253Calculus (Other Topics)Chapter 10 – Infinite SeriesSection 10.8 – Maclaurin and Taylor Series; Power SeriesCopyright © 2006 by Ron Wallace, all rights reserved.Taylor & Maclaurin PolynomialsFrom 10.9( )000( )( ) ( )!knknkf xp x x xk== -�200 0 0 0(3) ( )30 00 0''( )( ) '( )( ) ( )2( ) ( ) ( ) ( )6 !nnf xf x f x x x x xf x f xx x x xn= + - + -+ - + + -gggTaylor vs. Maclaurin? Maclaurin  x0 = 0Taylor & Maclaurin Series( )000( )( ) ( )!kknkf xp x x xk�== -�200 0 0 0(3) ( )30 00 0''( )( ) '( )( ) ( )2( ) ( ) ( ) ( )6 !nnf xf x f x x x x xf x f xx x x xn= + - + -+ - + + - +ggg gggMaclaurin x0 = 0NOTE: This assumes that f (n)(x0) exists for all n.Power Series2 30 1 2 3nnc c x c x c x c x= + + + + + +ggg gggNOTE: Every Maclaurin series is a power series.0kkkc x�=�where, each ck is a constant.Convergence Power Series0kkkc x�=�convergent?For what values of x isIs there a value of x for which any power series is convergent?Yes: 0x =0(0)kkkc�==�0cConvergence Power Series0kkkc x�=�convergent?For what values of x isFor any power series, one of the following will be true:a. The series only converges when x = 0.b. The series converges absolutely for all real values of x.c. The series converges absolutely for all x  (-R, R) and diverges when x > R or x < -R. The behavior a x =  R can vary.R is called the Radius of Convergence. (-R,R) or [-R,R] or (-R,R] or [-R,R) is called the Interval of Convergence.Power Series: Example1( )1f xx=-and determine its interval of convergence.Find the power series for( )10( ) 1 1xf x x-== - =( )4(3)0( ) 2 3 1 3!xf x x-== - =g( )20'( ) 1 1xf x x-== - =( )5(4)0( ) 2 3 4 1 4!xf x x-== - =gg( )30''( ) 2 1 2xf x x-== - =( )( 1)( )0( ) ! 1 !nnxf x n x n- +== - =Power Series: Example1( )1f xx=-and determine its interval of convergence.?Find the power series for( )( 1)( )0( ) ! 1 !nnxf x n x n- +== - =011kkxx�==-�1Ratio Test: lim 1Divergent if 1 (Why?)kkkxxxx+��= <=�Interval of Convergence is ( 1,1).\ -Power Series in x–x000( )kkkc x x�=-�2 30 1 0 2 0 3 00( ) ( ) ( ) ( )nnc c x x c x x c x xc x x= + - + - + -+ + - +ggg gggNOTE: Every Taylor series is a power series in x-x0.where, each ck is a constant.Convergence Power Series in x-x000( )kkkc x x�=-�convergent?For what values of x isFor any power series, one of the following will be true:a. The series only converges when x = x0.b. The series converges absolutely for all real values of x.c. The series converges absolutely for all x  (x0-R, x0+R) and diverges when x > x0+R or x < x0-R. The behavior a x = x0R can vary.R is called the Radius of Convergence. The set of all values of convergence is called the Interval of