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Unformatted text preview: x k ≤ x k-1 , it follows that-1 x k ≤ -1 x k-1 , and x k +1 = 2-1 x k ≤ 2-1 x k-1 = x k . Thus the sequence is monotonically decreasing. We next show that it is bounded below. Indeed, we show that x n > 1. We, again, use induction. The base case is given by hypothesis. Assuming that x k > 1, we examine x k +1 . Indeed, x k > 1 gives us that-1 x k >-1 and x k +1 = 2-1 x k > 2-1 = 1 which establishes the bound from below. Since the sequence is monotonically decreasing and bounded below, it must converge. Set L = lim n →∞ x n . Then, we must have that L = 2-1 L 1 or L 2 = 2 L-1 . Thus, L = 1. 2...
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- Spring '10
- Calculus, Mathematical analysis, Limit of a sequence, Xn, xk