lec 14 - Monday, January 30- Lecture 14 : Monotone...

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Monday, January 30 Lecture 14 : Monotone convergence (sequence) theorem (Refers to Section 8.1 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : State the Monotone sequence theorem ( Monotone convergence theorem ), apply the Monotone convergence theorem to determine convergence of a sequence. If we know a sequence converges but cannot compute its limit we can always approximate its limit by choosing a term in its “tail-end. A rough estimate of its level of accuracy can be obtained by comparing the value of a few terms around the term we have chosen. But before we approximate the limit of sequence we must determine whether the sequence converges at all. The Monotone sequence theorem provides a tool to do this. 14.1 Definitions A sequence of numbers { a i : i = 1, 2, 3,. ..} is said to be increasing if a i+ 1 > a i for all i 1. decreasing if a i+ 1 < a i for all i 1. nonincreasing if a i+ 1 a i for all i 1. nondecreasing if a i+ 1 a i for all i 1. A sequence of numbers is said to be monotonic or monotone if it is either increasing, decreasing, non-increasing or non-decreasing. 14.2 Definitions Let S = { a i : i = 1, 2, 3,. ..} be a sequence of numbers. We say that a number M is an upper bound of S if a i Μ for all i . We say that a number L is a lower bound of S is L a i for all i . If a sequence has both an upper bound and a lower bound then we say it is bounded . If a sequence has no upper (lower) bound we say that the sequence is unbounded above ( below ). The smallest upper bound of a sequence is called the " least upper bound " (or supremum ) and the largest upper bound of a sequence is called the " greatest lower bound " (or infimum ). Note that these definitions apply to arbitrary subsets of R as well as sequences.
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14.3 The Monotone sequence theorem ( Monotone convergence theorem ) Any bounded monotonic sequence must converge to some number. Proof: (
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lec 14 - Monday, January 30- Lecture 14 : Monotone...

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