Taylor and Maclaurin SeriesConvergent Power Series FormTaylor SeriesSlide 4Guidelines for Finding Taylor SeriesSeries for Composite FunctionBinomial SeriesSlide 8Combining Power SeriesAssignmentTaylor and Maclaurin SeriesLesson 9.10Convergent Power Series Form•Consider representing f(x) by a power series•For all x in open interval I•Containing c•Then ( )( )nnf x a x c= -�( )( )!nnf can=2"( )( ) ( ) '( )( ) ( ) ...2!f cf x f c f c x c x c= + - + - +Taylor Series•If a function f(x) has derivatives of all orders at x = c, then the seriesis called the Taylor series for f(x) at c.•If c = 0, the series is the Maclaurin series for f .( )20( ) "( )( ) ( ) '( )( ) ( ) ...! 2!nnnf c f cx c f c f c x c x cn�=- = + - + - +�Taylor Series•This is an extension of the Taylor polynomials from section 9.7•We said for f(x) = sin x, Taylor Polynomial of degree 72 374 5 6 7sin ( ) 0 02! 3!0 04! 5! 6! 7!x xx M x xx x x x� = + + � - +� + + � -Guidelines for Finding Taylor Series1. Differentiate f(x) several times•Evaluate each derivative at c2. Use the sequence to form the Taylor coefficients•Determine the interval of convergence3. Within this interval of convergence, determine whether or not the series converges to f(x)( )( ), "( ), '''( ),..., ( )nf c f c f c f c( )( )!nnf can=Series for Composite Function•What about when f(x) = cos(x2)?•Note the series for cos x•Now substitute x2 in for the x's2 4 6cos 1 ...2! 4! 6!x x xx = - + - +4 8 122cos 1 ...2! 4! 6!x x xx = - + - +Binomial Series•Consider the function•This produces the binomial series•We seek a Maclaurin series for this function•Generate the successive derivatives•Determine•Now create the series using the pattern( )( ) 1kf x x= +( )(0) ?nf =2"( )( ) (0) '(0) ...2!f cf x f f x x= + �+ � +Binomial Series• • We note that•Thus Ratio Test tells us radius of convergence R = 1•Series converges to some function in interval-1 < x < 1 2( 1) ( 1) ... ( 1)( ) (1 ) 1 ...2 !nkk k x k k k n xf x x kn� - � - �� - - �= + = + �+ + +1lim 1nnnaa+��=Combining Power Series•Consider•We know•So we could multiply and collect like terms ( ) arctanxf x e x= �2 33 51 ...2! 3!arctan ...3 5xx xe xandx xx x= + + + += - + -2 31arctan ...6xe x x x x� = + + +Assignment•Lesson 9.10•Page 685•1 – 29
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