e22 - counterexample(a If the sequence of partial sums of a...

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Calculus II, Exam 2 Redemption November 15, 2007 Name: 1. (8 points) (a) Explain why the series X n =1 ( - 1) n ln(ln(ln(100 n ))) converges. (b) Find N such that | s - s N | < . 001 Note: You are not asked to find the best such N . You must however give supporting evi- dence that your assertion is correct. 2. (10 points) (a) Explain why the series X n =1 1 n 2 + 2 n diverges. Is the sequence of partial sums unbounded? (b) Find N such that s N = N X n =1 1 n 2 + 2 n > 100 Note: You are not asked to find the best such N . You must however give supporting evi- dence that your assertion is correct. 3. (12 points) True or false. If true, briefly explain why. If false, show so with a simple
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Unformatted text preview: counterexample. (a) If the sequence of partial sums of a series is bounded then the series converges. (b) If ∑ a n converges then ∑ (-1) n a n converges. (c) Suppose ∑ ∞ n =0 a n , ∑ ∞ n =0 b n and ∑ ∞ n =0 a n b n converge. Then ( ∑ ∞ n =0 a n )( ∑ ∞ n =0 b n ) = ∑ ∞ n =0 a n b n (d) If | a n | ≥ | b n | for all n and ∑ b n diverges then ∑ a n diverges. 1...
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This note was uploaded on 10/18/2011 for the course MATH V1102 taught by Professor Mosina during the Fall '08 term at Columbia.

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