Math118_S10_W10

Math118_S10_W10 - Math118 (M. Kohandel) -Outline (week 10)...

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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 1 Outline (week 10) 10. 1 Absolute and conditional convergence, alternating series (10.12) 10. 2 Exact and approximate values of infinite series (10.13)
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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 2 10. 1 Absolute and conditional convergence, alternating series So far we only considered positive series. However, some series may contain both positive and negative terms - one important type is an alternating series (terms alternate in sign). In general, |) | ( all for 0 , ) 1 ( 0 3 2 1 0 ) 1 ( 0 3 2 1 0 n n n n n n a c n n c a n a a a a a a c c c c c n n n  Alternating series test (ATS): If an integrating series 0 ) 1 ( n n n a satisfies (i) 0 lim n n a and (ii) n n a a 1 for all n greater than some integer (i.e. the sequence } { n a decreases eventually), then the series is convergent. Example 10.1: Test the following series for the convergence or divergence. (a) 1 ) 1 ( n n n , (b) 1 1 2 ) 1 ( n n n n , (c) 1 3 2 1 1 ) 1 ( n n n n All of the given series are alternating, so we try to verify the conditions of AST. (a) First we note that 0 1 lim n n , and the sequence } 1 { n is decreasing - the series is convergent. (b) Since 0 2 1 1 2 lim n n n , the series is divergent. (c) We have 0 1 lim 3 2 n n n , so we check the second condition. To see whether the given series is decreasing or not, we use the related function 3 3 0 2 3 3 3 2 2 0 2 0 ) 1 ( ) 2 ( ) ( 1 ) (  x x x x x x f x x x f x The function is decreasing for 3 2 x . Thus the series is eventually decreasing. Both conditions are satisfied, so the given series is convergent. The following definition/theorem make the tests that we have discussed so far still useful for series with positive and negative terms.
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Math118 (M. Kohandel) ---------------------------------------------------------------------------------------------------------------------------- 3 Definition: A series 1 n n c is said to be absolutely convergent if the series of absolute values 1 | | n n c converges. Theorem: If a series is absolutely convergent, then it is convergent. Example 10.2: Consider the series (a) 1 2 1 ) 1 ( n n n . This series is convergent because the series of absolute values 1 2 1 n n is convergent (p-series with 2 p ). Definition: A series that converges but does not converge absolutely (i.e. 1 n n c converges but 1 | | n n c does not converge) is said to converge conditionally . Example 10.3: An example of conditionally convergent series is 1 ) 1 ( n n n . We know that this series converges; by AST: 0 ) / 1 ( lim n n and the sequence } / 1 { n is decreasing (in fact, the sum of this series is 2 ln , as we mentioned in the previous session). However, the series 1 1 n n diverges (harmonic series. or p-series with 1 p ).
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This note was uploaded on 02/06/2011 for the course MATH 118 taught by Professor Zhou during the Spring '08 term at Waterloo.

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Math118_S10_W10 - Math118 (M. Kohandel) -Outline (week 10)...

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