MTH 253 Calculus (Other Topics)PolynomialsPower Series: A polynomial with infinite degree.Power Series about x = aExample: A Geometric Power SeriesConvergence of Power SeriesSlide 7Slide 8Example …Slide 10Slide 11Slide 12Operations on Power SeriesMTH 253Calculus (Other Topics)Chapter 11 – Infinite Sequences and SeriesSection 11.7 – Power SeriesCopyright © 2009 by Ron Wallace, all rights reserved.Polynomials2 30 1 2 3( )nn nP x c c x c x c x c x= + + + + +ggg0( )nkn kkP x c x==�where, each ck is a constant and cn  0.Polynomial of Degree nPolynomials can be added, subtracted, multiplied, differentiated, and integrated.Power Series: A polynomial with infinite degree.2 30 1 2 3nnc c x c x c x c x= + + + + + +ggg ggg0nnnc x�=�where, each cn is a constant.Like polynomials… Power Series can be added, subtracted, multiplied, differentiated, and integrated.Power Series about x = a( ) ( ) ( )20 1 2nnc c x a c x a c x a= + - + - + + - +ggg ggg( )0nnnc x a�=-�where, each cn & a are constants.NOTE:• The value a is called the “center” of the power series.• When x = a, the power series is equal to c0.Example: A Geometric Power Series2 301 nnx x x x�== + + + +�gggNote that this is a geometric series with a = 1 and r = x.When does this series converge?What does it converge to?1x <11 x-That is …01, if 11nnx xx�== <-�Convergence of Power Series( )0nnnc x a�=-�convergent?For what values of x isIs there a value of x for which any power series is convergent?Yes: x a=0( )nnnc a a�=- =�0cConvergence of Power SeriesFor any power series, one of the following will be true:a. The series only converges when x = a.b. The series converges absolutely for all real values of x.c. The series converges absolutely for all x  (a-R, a+R) and diverges for x < a-R & x > a+R. The behavior at x = aR can vary.R is called the Radius of Convergence. (a-R,a+R) or [a-R,a+R] or (a-R,a+R] or[a-R,a+R) is called the Interval of Convergence.( )0nnnc x a�=-�convergent?For what values of x isConvergence of Power SeriesBasic Method:1. Use the ratio test (limit of the absolute value of the ratio of consecutive terms must be less than 1).2. Check the endpoints using some other test (using those two specific values of x).( )0nnnc x a�=-�convergent?For what values of x isExample …Determine the convergence of …0!nnxn�=�Radius of Convergence? Interval of convergence?1 of 4Example …Determine the convergence of …0( 1)n nnx�=-�Radius of Convergence? Interval of convergence?2 of 4Example …Determine the convergence of …( )03 2nnxn�=-�Radius of Convergence? Interval of convergence?3 of 4Example …Determine the convergence of …204 ( 1)nnnnxn�=+�Radius of Convergence? Interval of convergence?4 of 4Operations on Power Series( ) ( )If ( ) & ( )converge absolutely for , then ...n nn nf x r x a g x s x bx I= - = -�� �( ) ( )( ) ( )( )( )f x g xf x g xdf xdxf x dx��All convergeabsolutely for x I�These operations can be carried out “term-by-term”.I is the intersection of the intervals of convergence of f(x) and