74-hw8s - MATH 74 HOMEWORK 8 1. Suppose that an is...

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MATH 74 HOMEWORK 8 1. Suppose that a n is convergent and b n is not convergent. We are to show that a n + b n is not convergent. Suppose to the contrary that a n + b n is convergent. Since the constant sequence - 1 is convergent (Theorem 29 in the class notes), and a n is convergent, the sequence - 1 · a n = - a n is convergent (Theorem 31). We then conclude from Theorem 30 that the sequence ( a n + b n ) + ( - a n ) = b n , being the sum of convergent sequences, is convergent. This is a contradiction. 2. If a n is a convergent sequence and b n is not convergent, we cannot prove that a n b n is not convergent without additional assumptions. For example, let a n = 1 n and let b n = n . We showed in class that a n was convergent, and you can show (using an argument similar to the one for n on April 4) that b n is not convergent. But the sequence of products a n b n is the constant sequence 1, which is convergent (Theorem 29). (One can prove: if a n is a convergent sequence of nonzero numbers with nonzero limit L , and b n does not converge, then a n b n does not converge. It goes a lot like problem 1.) 3. Suppose that the sequence b n is convergent, say to the number L . Since the constant sequence 6 converges (Theorem 29) it follows from Theorem 30 that the sequence 6 + b n also converges (to 6 + L ). Note (or prove by induction) that
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This homework help was uploaded on 04/02/2008 for the course MATH 74 taught by Professor Courtney during the Fall '07 term at University of California, Berkeley.

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74-hw8s - MATH 74 HOMEWORK 8 1. Suppose that an is...

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