exam2solns - Exam 2 Solutions 1. False. For example, the...

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Exam 2 Solutions 1. False. For example, the odd terms of the alternating harmonic series X n =1 ( - 1) n - 1 n form the series X k =1 1 2 k - 1 which diverges by the Integral Test. 2. (i) s 1 = 1 2 , s 2 = 5 6 , s 3 = 23 24 , and s 4 = 119 120 . It seems that s n = ( n +1)! - 1 ( n +1)! = 1 - 1 ( n +1)! . (ii) Base step holds: 1 - 1 (1+1)! = 1 - 1 2 = 1 2 = s 1 . Now assume s m = 1 - 1 ( m +1)! for some m 1. Then we have s m +1 = s m + m + 1 ( m + 1 + 1)! = ± 1 - 1 ( m + 1)! ² + m + 1 ( m + 2)! = 1 - 1 ( m + 2)! and the induction proof is complete. (iii) Since s n = 1 - 1 ( n +1)! for all n 1, it’s clear that s n 1 as n → ∞ . Hence X n =1 n ( n + 1)! = 1. 3. (i) Diverges: the even terms of the series converge to 1 and therefore the sequence of terms doesn’t converge to 0. (ii) Converges: use the Integral Test. (iii) Converges: use the Ratio Test. (iv) Converges: use the Root Test. 4. First note that
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This note was uploaded on 04/09/2010 for the course MATH 312 taught by Professor Wynne during the Fall '09 term at Simons Rock.

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exam2solns - Exam 2 Solutions 1. False. For example, the...

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