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mat25-hw7solns - Math 25 Advanced Calculus UC Davis Spring...

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Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 — Solutions to Homework Assignment #7 1. Prove that the sequence a n = 1 · 3 · 5 · · · (2 n - 1) 2 · 4 · 6 · · · (2 n ) converges. Proof. We will apply the monotone convergence theorem. Note that since 2 n - 1 2 n < 1 we have that a n +1 < a n . So the sequence { a n } is decreasing monotonically. Clearly, the sequence is bounded below by zero. So, by the monotone convergence theorem, it must converge. 2. Define a sequence { x n } by x 1 = 3 , x 2 = q 3 + 3 , x n +1 = 3 + x n . Prove that the sequence converges and find its limit. For a small bonus credit, answer the same question when 3 is replaced an arbitrary integer k 2. Proof. We show that the sequence converges by applying the monotone convergence theorem. We will proceed by induction. First, we claim that x n < 3 for all n . Note that 3 < 3. Now, suppose that x n < 3. Then, x n +1 = 3 + x n 3 + 3 = 6 < 3 , which shows the the sequence x n is bounded above.
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