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3100_Homework9_sols

# 3100_Homework9_sols - SOLUTIONS TO PROBLEM SET 9 PETE L...

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SOLUTIONS TO PROBLEM SET 9 PETE L. CLARK All problems except the last are taken from § 2 . 3 of our text. 1) Let { s n } be a sequence and { c n } and { d n } be subsequences given by c n = s 2 n and d n = s 2 n +1 . Assume that lim n →∞ c n = lim n →∞ d n = L . Show that lim n →∞ s n = L . Solution: Let ϵ > 0. Since the sequences { c n } and { d n } each converge to L , there exists N 1 N such that for all n N 1 , | c n - L | < ϵ and N 2 N such that for all n N 2 , | d n - L | < ϵ . Let N = 2 max( N 1 , N 2 ) + 1. Any n N may be written as 2 k or 2 k + 1 with k max( N 1 , N 2 ). If n = 2 k , then | s n - L | = | s 2 k - L | = | c k - L | < ϵ, whereas if n = 2 k + 1, then | s n - L | = | s 2 k +1 - L | = | d k - L | < ϵ. Thus s n L . Remark: More generally if we partition the natural numbers N into any finite collection of infinite sets S 1 . . . S K , then a sequence s n L iff for all 1 i k , the subsequence obtained by restricting the terms to lie in S i converges to L . 2) What does the ratio test tell you about the following series? a) n 2 n n ! . Solution: The ratio test limit is ρ = lim n →∞ 2 n +1 ( n + 1)! n ! 2 n = lim n →∞ 2 n + 1 = 0 < 1 , so the series converges. b) n n ! 2 n . Solution: Since the terms of this series are the reciprocals oft he terms of the series in part a), the ratio test limit ρ will be the reciprocal of the ρ for part a), i.e., ρ = 1 0 + = . So the series diverges. c) n n 3 n . c Pete L. Clark, 2011. 1

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2 PETE L. CLARK Solution: The ratio test limit is ρ = lim n →∞ n + 1 3 n +1 3 n n = lim n →∞ 1 3 n + 1 n = 1 3 < 1 , so the series converges. d) n 1 n 2 . Solution: The ratio test limit is ρ = lim n →∞ n 2 ( n + 1) 2 = 1 , so the ratio test fails , i.e., it does not tell us whether the series converges. In fact this is a p -series with p = 2 > 1, so by other means – the Condensation Test or the Integral Test – we know the series converges. So it is not the convergence of the series that is failing, it’s the Ratio Test! 3) Which of the following series are absolutely convergent, nonabsolutely conver- gent or divergent? a) n ( - 1) n n 2 . Solution: The associated series of absolute values is the convergent p -series n 1 n 2 , so the given series is absolutely convergent.
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