242hw7solns - Math 242 Homework 7 Solutions In questions 1...

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Solutions In questions 1 through 8, determine whether the series converges or diverges. (Give reasons.) 1. X n =1 1 2 n Solution. This series is 1 2 times the harmonic series, so it diverges . 2. X n =1 ln ± n 2 + 1 2 n 2 + 1 ² Solution. Note that when n → ∞ , we have n 2 + 1 2 n 2 + 1 1 2 . Therefore the terms of this series approach ln ± 1 2 ² , which is not 0. Therefore, by the “Pre-Test” (or “Test for Divergence”), the series diverges . 3. X n =1 1 + 3 n 2 n Solution. This series is the sum of the two series X n =1 1 2 n and X n =1 3 n 2 n = X n =1 ± 3 2 ² n . The second of these is a geometric series with r = 3 2 , so it diverges. Therefore the original series diverges . 4. X n =1 n e n Solution A. We can say n < 2 n . Therefore n e n < 2 n e n = ± 2 e ² n . But X ± 2 e ² n is a geometric series with r = 2 e < 1, which converges. Therefore by basic comparison, the original series converges . 1
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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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242hw7solns - Math 242 Homework 7 Solutions In questions 1...

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