exercises_8 - Analysis 1: Exercises 8 1. Let p > 0, a1...

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Unformatted text preview: Analysis 1: Exercises 8 1. Let p > 0, a1 > 0. Let (an )n∈N+ be a sequence defined recursively by (∀n ∈ N+ ) an+1 = 1 2 an + p an . Prove that (an )n∈N+ converges and compute its limit. 2. Prove convergence of the series below. ∞ 1 (a) . (3n − 1)(3n + 2) n=1 ∞ (b) √ √ √ ( n + 2 − 2 n + 1 + n). n=1 3. Investigate the convergence of the series below. ∞ n . (a) 2n − 1 n=1 ∞ (b) (c) 1 √. n n=1 ∞ n=1 ∞ (−1)n n . n+1 √ (d) (e) 1 . n n+1 n=1 ∞ n=1 3n . n5n ∞ ∞ 4. (a) The series of positive terms converges. (b) The series n=1 n=1 ∞ ∞ an converges. Prove that the series n=1 ∞ a2 n a2 , n n=1 b2 converge. Prove that the series n n=1 |an bn | converges. 5. Use Cauchy’s or d’Alembert’s test to investigate whether or not the series below converge. ∞ (a) n=1 ∞ 100n . n! (b) (c) √ (n n − 1)n . n2−n . n=1 ∞ n=1 ...
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This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Winter '10 term at Bristol Community College.

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