lect118_14_w13 - Friday February 1 Lecture 14 Monotone...

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Friday, February 1±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 theoremprovides a tool to do this. 14.1 Definitions ±A sequence of numbers {ai:i = 1, 2, 3,...} is said to be increasingif ai+1!aifor all i t1. decreasingif ai+1²aifor all i t1. nonincreasingif ai+1daifor all i t1. nondecreasingif ai+1taifor all i t1. A sequence of numbers is said to be monotonicor monotoneif it is either increasing, decreasing, non-increasing or non-decreasing. 14.2Definitions ±Let S= {ai:i= 1, 2, 3,...} be a sequence of numbers. We say that a number M is an upper bound of S if aid0for alli. We say that a number L is a lower bound of SisLdaifor alli.

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