Power SeriesDefinitionConvergence of Power SeriesExampleDealing with EndpointsSlide 6Try AnotherPower AssignmentPower SeriesLesson 8.7Definition•A power series in (x – c) has the form•Consider this as an extension of a polynomial in x( ) ( ) ( )20 1 11...kkka x c a a x c a x c�=- = + � - + � - +�Convergence of Power Series•For the power seriesexactly one of thefollowing is true1. The series converges for all x2. The series converges only for x = 03. The series •converges absolutely for all x in (-R, R)•diverges for |x|> R•may converge or diverge at R or -R 0kkka x�=��Example•Consider the power series•What happens at x = 0?•Use generalized ratio test for x ≠ 0•Try this1!kkxk�=�( )11 !lim ?!kkkxkLxk+��+= =1!kkk x�=�Dealing with Endpoints•Consider •Converges trivially at x = 0•Use ratio test•Limit = | x | … converges when | x | < 1Interval of convergence -1 < x < 111kkk x�=- ��1lim ?1kkkk xk x+���=- �Dealing with Endpoints•Now what about when x = ± 1 ?•At x = 1, diverges by the divergence test•At x = -1, also diverges by divergence test•Final conclusion, convergence set is (-1, 1)11 1kkk�=- ��( )11 1kkk�=- �-�Try Another•Consider•Again use ratio testShould get which must be < 1 or -1 < x < 5 •Now check the endpoints, -1 and 5 ( )2123kkkk x�=-�123L x= -Power Assignment•Lesson 8.7•Page 552•Exercises 1 – 25
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