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242hw10solns - Math 242 Homework 10 Solutions ex 1 dx as a...

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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 0 . 2 0 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
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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 0 . 2 0 1 1 + x 5 dx = Z 0 . 2 0 1 - x 5 + x 10 - x 15 + · · · dx = x - x 6 6 + x 11 11 - x 16 16 + · · · 0 . 2 0 = 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 + 1 The sum of just the first two terms is 0 . 2 - (0 . 2) 6 6 = 0 . 2 - 0 . 000064 6 0 . 2 - 0 . 0000106667 = 0 . 1999893333 . When I enter the command int(1/(1+x^5),x=0..0.2); into Maple, it gives me 0 . 1999893352.
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