242hw10solns - Math 242 Homework 10 Solutions 1 Express the...

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Unformatted text preview: Math 242 Homework 10 Solutions 1. Express the indefinite integral Z e x- 1 x dx as a power series. e x = 1 + x 1! + x 2 2! + x 3 3! + x 4 4! + ··· = ∞ X n =0 x n n ! e x- 1 = x 1! + x 2 2! + x 3 3! + x 4 4! + ··· = ∞ X n =1 x n n ! e x- 1 x = 1 1! + x 2! + x 2 3! + x 3 4! + ··· = ∞ X n =1 x n- 1 n ! Z e x- 1 x dx = x 1 · 1! + x 2 2 · 2! + x 3 3 · 3! + x 4 4 · 4! + ··· = ∞ X n =1 x n n · n ! 2. Express the definite integral Z . 2 1 1 + x 5 dx as an infinite series. Then find the sum of the first two terms of that series. Compare that to the value of the integral that Maple gives you. 1 1 1- x = 1 + x + x 2 + x 3 + ··· = ∞ X n =0 x n 1 1 + x = 1- x + x 2- x 3 + ··· = ∞ X n =0 (- 1) n x n 1 1 + x 5 = 1- x 5 + x 10- x 15 + ··· = ∞ X n =0 (- 1) n x 5 n Z . 2 1 1 + x 5 dx = Z . 2 1- x 5 + x 10- x 15 + ··· dx = x- x 6 6 + x 11 11- x 16 16 + ··· . 2 = 0 . 2- (0 . 2) 6 6 + (0 . 2) 11 11- (0 . 2) 16 16 + ··· = ∞ X n =0 (- 1) n (0 . 2) 5 n +1 5 n...
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This note was uploaded on 12/07/2011 for the course MATH 242 taught by Professor Wang during the Spring '08 term at University of Delaware.

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242hw10solns - Math 242 Homework 10 Solutions 1 Express the...

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