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This 32 page Class Notes was uploaded by Claudine Friesen on Monday September 28, 2015. The Class Notes belongs to MATH104 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 17 views. For similar materials see /class/215402/math104-university-of-pennsylvania in Mathematics (M) at University of Pennsylvania.
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Date Created: 09/28/15
L 7 L a U131 WT 771 515my Infinite Series Eat You get when M 779 4 U H U Q 1 kidz dj n minimEmmy k 1rd 3 1 w 7 H H Hf 1335 m v X W N r 39 f s V y 39ra y i h i m d wzgmgeag in Hf M 21 ometric Seri it Um n im 515d I 3 39 5 M 3w 0 My g2 w H RN fa 441 23 19 IJ L j if 1 D J 3 F3 jg ja 39 WU harmonic Serie if M O quot 7 0 inn u L T M i x rties of Li quot it 39 k V firr A wig WISERV I h 7 LiaAg V quotq UM r H y A n H 7 IL 7 g L 7 J K 9 7 6111fm ssw s m imam 9 i Z i W 39 9 51A L J W Math 104006 Chapter 128 Power Series Outline For Today 0 Power Series Power Series A power series is a series of the form 00 2 n 2 n0 Where x is a variable and the ci are the coefficients of the series Power Series Continued A power series may converge for some values of x and diverge for others The sum ofthe series is a function fx chxquot CO clxc2x2 n20 Whose domain is the collection of x for which the series converges Example Con der fxzxquot 1xx2 n20 We know this converges if and only if 1 lt x lt 1 Further in this case fx 11 x Example 00 x72 If fx 207 what is the domain of fx AAX D1ltxSl B1ltxlt1 E1Sx 1 C 1 S x lt 1 F None ofthe above Example 39 If fx what is the domain of fx AAx D1ltxs1 B1ltxlt1 E1 x 1 F None of the above Power Series in xa 0 A quotpower series in x aquot or quota power series centered at a or quota power series about a is of the form chx aquot CO clx a62x a2 n20 Radius of Convergence 0 For any given power series Zen x a 720 0 There are only three options for the radius of convergence Radius of Convergence Continued 0 Radius of Convergence is 0 0 In this case the series ch x a 720 converges when xa and diverges everywhere else Radius of Convergence Continued Radius of Convergence is a real number R 0 In this case the series ch x a 720 converges if x a lt R and diverges if x a gt R Radius of Convergence Continued Radius of Convergence is oo 0 In this case the series ch x a 720 converges for all real x Interval of Convergence 0 For any given power series Zen x a 720 0 The interval of convergence is the domain of the function There are only six options for the interval of convergence Interval of Convergence Radius is 0 lfthe radius of Convergence of 2 x aquot 720 is 0 0 Then the interval of convergence is a a Interval of Convergence Radius is oo 0 lfthe radius of Convergence of 2 x aquot 720 is 00 Then the interval of convergence is oo oo Interval of Convergence Radius is R Ifthe radius of Convergence of 2 x a 720 is R 0 Then there are four possible interval of convergences R a a R R a a R R a a R R a a R Example 0 What is the interval of convergence of 274196quot n O A 1 1 D 00 B 1 O E oo 00 C 1 O F None ofthe above Example What is the interval of convergence of Z nixquot n0 A 1 1 B 1 0 E oo 00 C 1 O F None of the above Example oo 71 X 0 What is the interval of convergence of Z n n20 A 1 1 D 00 B 1 O E oo 00 C 1 O F None ofthe above Exa mple What is the interval of convergence of Z x n0 n n A 1 1 D 0 0 F None of the above B 1 O C 1 O