midterm 2 take home

Payment schedule a the amount of payment on the

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Unformatted text preview: rsity X account. Payment Schedule A. The amount of payment on the payment # n is exactly 1 . one n-th of a dollar =$ n See Table 1. B. As for the timing of payments, the interval between two consecutive payments is exactly one half of the previous interval. See Table 2. C. The payment will conclude 48 hours after the payment # 1, but not before. payment # amount #1 $1 #2 $ 1 /2 #3 $ 1 /3 #4 $ 1 /4 #5 $ 1 /5 #6 $ 1 /6 . . . . . . Table 1. 2 payment # timing of payment #1 −− #2 24 hours after payment #1 #3 24/2 hours after payment #2 #4 24/4 hours after payment #3 #5 24/8 hours after payment #4 #6 24/16 hours after payment #5 #7 24/32 hours after payment #6 . . . . . . Table 2. Suppose you are Madam Rich’s financial advisor. Your job is not to let her lose all her money. Should you tell her that it is okay to sign? Explain. 3 [III] (Take-home; 20pts) fx gx (1a) = = −1 2 For x with −2 < x < 2, define ∞ ln n=2 x2 n2 1− n , 1 x+ lim − k n− ∞ → k=2 n 1 2 k+x . k−x ln k=2 Agree that f x can be written as −1 lim ln 2 n− ∞ → 1+ x 2 x 2 1− x 3 1+ ··· · 1+ x n 1− 1− x n x 3 · . Agreed. (1b) Agree that g x can be written as n 1 1 lim − x+ ln k 2 n− ∞ → k=2 1+ x 2 1− x 2 Agreed. 4 1+ x 3 · ··· · 1 + x n 1− x 3 · ··· · 1 − x n . (2a) Give the Taylor series expression of −f x Answer for (2a) ∞ m=2 −1 . : ζm −1 m m−1 −1 x −1 ζ + x xm + − ζ =− −1 ζ x − −1 ζ x −1 + ··· . ζ x : Work for (2a) −f x +gx +gx = lim − n− ∞ → n k=2 = lim − n− ∞ → n k=2 1 x x + ln 1 + k 2 1 x+ k n ln 1+ k=2 5 1+ x k x 3 · ··· · 1+ x n = n 1 lim − x+ k n− ∞ → k=2 n = lim n− ∞ → k=2 n = lim n− ∞ → k=2 ∞ = k=2 − − −1 x k x k −1 = k=2 + −1 ∞ + −1 + m=2 m=2 −1 ∞ = m=2 −1 m−1 m−1 x k x k m−1 x k two Σs were interchanged k=2 x m−1 m−1 k=2 ∞ x k ∞ = x k m−1 m=1 x k m−1 m=1 −1 ∞ m=2 ∞ m=2 n ...
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This note was uploaded on 01/12/2014 for the course MATH 116 taught by Professor Scholle,minho during the Fall '08 term at Kansas.

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