m25-hw6-solutions

m25-hw6-solutions - Math 25 Advanced Calculus Fall 2010...

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Unformatted text preview: Math 25: Advanced Calculus Fall 2010 Homework # 6 – Solutions 6.1: Decide if each of the following sequences { a n } ∞ n =1 converges or diverges. If the sequence converges, state its limit. In either case, you must use the appropriate definition prove that the sequence converges to the desired limit or that the sequence diverges. (a). a n = 1 n 2 Converges: lim n →∞ 1 n 2 = 0. We know that lim n →∞ 1 n = 0, and by the product formula, lim n →∞ 1 n 2 = 0 2 = 0. (b). a n = n 2 + n n 2 Converges: lim n →∞ n 2 + n n 2 = 1. We can write n 2 + n n 2 = 1 + 1 n . By the sum rule, lim n →∞ 1 + 1 n = 1 + 0 = 0. (c). a n = n 2 + n n Diverges to infinity. We can rewrite n 2 + n n = n +1. For any M > 0, there is a natural number N > M by the Archimedean Principle. For any n ≥ N , n + 1 > N > M , and hence { n + 1 } diverges to infinity. (d). a n = cos( nπ ) Diverges. The sequence { a n } = { (- 1) n } , which we know to be divergent....
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This note was uploaded on 03/18/2012 for the course MAT MAT 25 taught by Professor Stevenklee during the Fall '10 term at UC Davis.

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m25-hw6-solutions - Math 25 Advanced Calculus Fall 2010...

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