Lecture 20 (Alternating Series Test)

Lecture 20 (Alternating Series Test) - Thursday, February...

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Unformatted text preview: Thursday, February 10 Lecture 20 : Alternating series (Refers to Section 8.2 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to : Define an alternating series , state the alternating series test , apply the alternating series test . 20.1 Definition An alternating series is a series of the form where c j is non-negative for all j . (In practice, the series need not start with j = 1. It can also start with j equal to any positive integer.). Also the series need not necessarily start with a positive term. One normally calls on the Alternating series test to test such series for convergence. It is described and proven below : 20.2 Theorem The alternating series test . ( AST ) If { c j } is a sequence of non-negative numbers such that 1) lim j c j = 0 and 2) the terms of { c j } are strictly decreasing, then the alternating series converges. Proof: Suppose lim j c j = 0 and { c j } is strictly decreasing. Let { S n : n = 1 to } be the partial sums of the series. First observe that - The sequence { S 2 n : n = 1 to } is a monotone increasin g sequence of positive terms and that it is bounded above...
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Lecture 20 (Alternating Series Test) - Thursday, February...

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