Homework_Set_8

# Homework_Set_8 - ··· = π 4 (2) [25 points] (a) Assuming...

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Math 1a B. Simon Fall 2005 HOMEWORK SET 8 DUE AT 1:00 PM ON MONDAY, NOVEMBER 21, 2005 (1) [50 points] Suppose a n > a n +1 0 for all n . DeFne f ( x ) = s n =0 ( - 1) n a n x n (a) Prove that this series is absolutely convergent for | x | < 1. Suppose from now on that a n 0 as n → ∞ . That means n =0 ( - 1) n a n is convergent. The point of the remaining parts is to prove that lim x 1 f ( x ) = s n =0 ( - 1) n a n L ( * ) (b) Let f k ( x ) be the partial sum used to deFne f ( x ) where n runs from 0 to k . Prove that f 2 m ( x ) f ( x ) f 2 m +1 ( x ) (c) Given ε , Fnd N so n N ⇒ | f n (1) - L | < ε/ 2, and then δ so 1 - δ < x < 1 implies | f N + j ( x ) - f N + j (1) | < ε/ 2 for j = 0 , 1. Prove that if 1 - δ < x < 1, then | f ( x ) - L | < ε and conclude that ( * ) holds. (d) What is n =1 ( - 1) n +1 /n ? (e) Prove that 1 - 1 3 + 1 5 - 1 7

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Unformatted text preview: ··· = π 4 (2) [25 points] (a) Assuming the value of the Gaussian integral i ∞-∞ e-x 2 2 dx = √ 2 π Fnd i ∞ √ x e-x dx (b) Prove that i ∞ x n + 1 2 e-x dx = (2 n + 1)! n ! 2 2 n +1 √ π for n = 0 , 1 , 2 , . . . . 1 2 HOMEWORK SET 8 DUE AT 1:00 PM ON MONDAY, NOVEMBER 21, 2005 (3) [25 points] Use Newton’s method to try to solve x = e-x starting with the guess x 1 = 0. What is the algorithm for x n +1 in terms of x n ? Use some program to implement the algorithm and Fnd the solution to 10-place accuracy....
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## This note was uploaded on 03/30/2010 for the course MA 1a taught by Professor Borodin,a during the Fall '08 term at Caltech.

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Homework_Set_8 - ··· = π 4 (2) [25 points] (a) Assuming...

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