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74-hw8 - x n is a convergent sequence of nonnegative...

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MATH 74 HOMEWORK 8 - DUE MONDAY, APRIL 9 In all of the following, you should be quite explicit when you use theorems. I recommend referencing them by numbers (see the lecture notes, especially April 2) or by suggestive names (eg “the sum rule for limits”). 1. Suppose a n and b n are sequences, and that a n is convergent (to a limit a ), and that b n is not convergent. Prove that a n + b n is not convergent. [Hint: use proof by contradiction, and the theorems from April 2.] 2. Can you also prove, under the same hypotheses as problem 1, that a n b n is not convergent? Why or why not? 3. Define a sequence b n by setting b 1 = 6 and b n +1 = 6 + b n for all n > 1. Thus b 1 = 6 , b 2 = 6 + 6 , b 3 = 6 + 6 + 6 , and so on. Assuming that the sequence b n is convergent, what is its limit? [Justify your calculation using the theorems from April 2. You may also need the following theorem, which you should refer to as “Theorem X” in your homework: If
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Unformatted text preview: x n is a convergent sequence of nonnegative numbers with limit L , then √ x n is a convergent sequence with limit √ L .] 4. Same question for c n given by c 1 = 1 and c n +1 = 1 1+ c n for n > 1. [Justify your calculation using the theorems from April 2.] 5. Define the sequence a n by a 1 = 2 and a n +1 = 2 a n +2 a n +2 for n > 1. It might help to do 5(c) first. 5(a). Prove that the sequence a n is bounded. (If you are starting this homework before I have defined it in class: exhibit a number B and a proof that | a n | ≤ B for all n ∈ N .) 5(b). Prove that the sequence a n is decreasing. (If you are starting this homework before I have defined it in class: prove that a n-a n +1 > 0 for all n . 5(c). If the sequence converges to a limit, what is the limit? 1...
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