L24 Power Series - Lecture 24 Power Series Given a sequence...

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Lecture 24: Power Series Given a sequence f a n g , the series 1 X n =0 a n x n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + ±±± is called a power series of variable x . The num- bers a n are called the coe±cients of series. More generally, 1 X n =0 a n ( x ² a ) n ; is called a power series centered at a , or a power series about a (We adopt the convention that ( x ² a ) 0 = 1 ) ex. The following series are called power series . 1 X n =0 x n ; 1 X n =0 x n n ! ; 1 X n =0 x n n 1
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Whether a power series converges or not, it de- pends on the choice of x . A power series always converges for x = 0 to the number a 0 . Question: For which x will the power series converge? 2
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Convergence of Power Series We use Ratio Test or Root Test to determine when would these power series converge. 1. 1 X n =1 x n n 2. 1 X n =0 n ! x n 3. 1 X n =0 x n n ! 3
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Radius and Interval of Convergence Interval of Convergence Radius of Convergence ( ±1 ; 1 ) R = 1 10 g ; f 0 g ; f ± g ; ²²² etc R = 0 ( ± 1 ; 11] ; [ ± 1 ; 11) ; [ ± 1 ; 11] ; ( ± 1 ; 11) R = 6 In other words, for a given power series 1 X n =0 a n ( x ± a ) n = a 0 + a 1 ( x ±
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