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Unformatted text preview: Sample Problems for Exam Three 1. Define the sum ∑ ∞ n =0 a n of an infinite series and show that if the sum exists, then lim n →∞ a n = 0. 2. Show that ∑ ∞ n =0 r n = 1 / (1 x ) for  x  < 1. 3. Use the Cauchy Criterion to show that if ∑  a n  converges, then ∑ a n converges. 4. Show that if ∑ a n = S , then ∑ ca n = cS for any constant c . 5. (a) Show that for a nonnegative series, ∑ a n converges if and only if the partial sums { S n } are bounded. (b) Show that if ∑ a n converges absolutely and { b n } is bounded, then ∑ a n b n converges absolutely. 6. Suppose that ∑ a n and ∑ b n are two nonnegative series and for some N and all n > N , a n ≤ b n . Show that (a) If ∑ b n converges, then ∑ a n converges; (b) If ∑ a n diverges, then ∑ b n divverges; 7. State and prove the Limit Comparison Test. 8. State and prove the Integral Comparison Test....
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This note was uploaded on 01/22/2012 for the course MAA 4103 taught by Professor Staff during the Fall '08 term at University of Florida.
 Fall '08
 Staff

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