mat25-hw7 - Math 25 Advanced Calculus UC Davis Spring 2011...

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Unformatted text preview: Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 — Homework Assignment #7 Homework due: Tuesday 5/17/11 at beginning of discussion section Reading material. Read sections 2.9, 2.10 in the textbook. Problems 1. Prove that the sequence a n = 1 · 3 · 5 · ... · (2 n- 1) 2 · 4 · 6 · ... · (2 n ) converges. (Note: you are not asked to find what the limit of the sequence is.) 2. Define a sequence ( x n ) ∞ n =1 by x 1 = √ 3 , x 2 = q 3 + √ 3 , x 3 = r 3 + q 3 + √ 3 , ... , x n +1 = √ 3 + x n , ... Prove that the sequence ( x n ) ∞ n =1 converges, and find its limit. For a small bonus credit, answer the same question for the more general sequence ( y n ) ∞ n =1 defined by y 1 = √ k, y n +1 = p k + y n , where k is an arbitary integer ≥ 2 (note that in that case the limit is a function of k ). 3. Define the sequence ( a n ) ∞ n =1 by a n = ( 1 + 1 n ) n . (a) Show that ( a n ) is increasing and bounded from above (see Example 2.36 in section 2.10 of the textbook for ideas) and deduce that it converges.in section 2....
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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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mat25-hw7 - Math 25 Advanced Calculus UC Davis Spring 2011...

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