HH_Sec_9_5 - R is 3. Example 4 What is the radius of...

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(9/1/08) Math 10B. Lecture Examples. Section 9.5. Power series and intervals of convergence Example 1 Find the radius of convergece of the power series s j =1 ( - 1) j +1 j x j = x - 1 2 x 2 + 1 3 x 3 - 1 4 x 4 + · · · . Answer: [Radius of convergence] = 1 (Figure A1a shows the partial sums for x = 0 . 75, where the series converges, and Figure A1b shows the partial sums for x = 1 . 2, where the series diverges.) N 10 20 y 1 ( x = 0 . 75) N 10 20 y 1 ( x = 1 . 2) y = N s n =1 ( - 1) n +1 n (0 . 75) n y = N s n =1 ( - 1) n +1 n (1 . 2) n Figure A1a Figure A1B Example 2 What is the radius of convergence of s n =0 1 (2 n )! x n ? Answer: The radius of convergence R is . Example 3 Find the radius of convergence of s n =1 x n n 3 3 n . Answer: The radius of convergence
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Unformatted text preview: R is 3. Example 4 What is the radius of convergence of s n =0 n ! x 2 n ? Answer: The radius of convergence R is 0. Interactive Examples Work the following Interactive Examples on Shenks web page, http//www.math.ucsd.edu/ashenk/: Section 10.7: Examples 14 Lecture notes to accompany Section 9.5 of Calculus by Hughes-Hallett et al. The chapter and section numbers on Shenks web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. 1...
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