extra credit hw viii

# extra credit hw viii - Seat Math 116 CALCULUS II EXTRA...

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Math 116 CALCULUS II EXTRA CREDIT HOMEWORK – VIII November 21 (Thu), 2013 Due date: December 5 (Thu), 2013 Seat # : Instructor: Yasuyuki Kachi Line #: 23590. ID # : Name : Problem 1. (0) Write out the following power series in an ‘honest’ form: I ( x ) = s n =0 p - 1 P n p 2 n P ! ! p 2 n - 1 P ! ! x n = 1 - 2 1 x + 4 · 2 3 · 1 x 2 - · · · · x 3 + · · · · · · x 4 - · · · · · · · · x 5 + ··· . 1

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(1) Find the radius of convergence R for the above power series. R = . Work for (1). Set a n = p - 1 P n p 2 n P ! ! p 2 n - 1 P ! ! . Then v v a n v v = , v v a n +1 v v = . = R = lim n -→ v v a n v v v v a n +1 v v = lim n = lim n p Pp P p P = lim n = . 2
(2) Decide whether the infnite series obtained by substituting x = 1 into the above power series is convergent or divergent. b Answer B : s n =0 p - 1 P n p 2 n P ! ! p 2 n - 1 P ! ! = 1 - 2 1 + 4 · 2 3 · 1 - 6 · 4 · 2 5 · 3 · 1 + 8 · 6 · 4 · 2 7 · 5 · 3 · 1 - 10 · 8 · 6 · 4 · 2 9 · 7 · 5 · 3 · 1 + 12 · 10 · 8 · 6 · 4 · 2 11 · 9 · 7 · 5 · 3

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