lect118_21_w13 - Friday February 15 Lecture 21 Absolute...

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Friday, February 15 Lecture 21 : Absolute convergence and conditional convergence. (Refers to Section 8.5 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to: Define "converges absolutely" and "converges conditionally”,s determine when a series converges absolutely and when it converges conditionally,. We have seen that the presence of negative terms in a series of numbers impacts on the convergence properties of that series. For example, the alternating harmonic seriesconverges but not the harmonic series. -What if we are given an alternating series whose terms are not decreasing and so the ASTdoes not apply? In this case we say that the AST“fails”. This means that this tool “fails to provide any information on the convergence properties of this series”. -What if we are given a series with negative terms but this series is not an “alternating series”? Again the ASTfails, since it cannot be used to determine the convergence properties of this series. -The following theorem will provide us with a tool that can be useful in such cases. 21.1 Theorem (Absolute convergence test) Suppose is a series where some of the an’s are possibly negative. Then Σn= 1 to | an | converges

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