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 na n +1 > 0 for all n . 5(c). If the sequence converges to a limit, what is the limit? 1...
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 Fall '07
 COURTNEY
 Math, sequence bn

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