Calculus II Notes 12.5 - Calculus II-Stewart Dr. Berg...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 12.5 12.5 Alternating Series Definition An alternating series is one in which consecutive terms have opposite sign. Theorem The Alternating Series Test Let a k { } be a sequence of positive numbers. If a k a k + 1 for all k N and lim k →∞ a k = 0 , then ( 1) k a k k = 0 converges. Outline of Proof: The even partial sums are all non-negative and decreasing to some limit by the monotone convergence theorem. The odd partial sums must converge to the same limit since the difference between the even and odd partial sums goes to zero. Examples A 1 ( ) k 1 k k = 1 is convergent since it is alternating and 1 k = a k a k + 1 = 1 k + 1 > 0 and lim k →∞ a k = lim k →∞ 1 k = 0 . Example B 1 ( ) k 1 k k + 1 k = 1 is divergent by the basic divergence theorem since a k = k k + 1 1 0 . Example C 1 ( ) k k ln k is convergent since it is alternating and 1 k ln k = a k a k + 1 = 1 k + 1 ( ) ln k + 1 ( ) > 0 and lim k →∞ a k = lim k →∞ 1 k + 1 ( ) ln k + 1 ( ) = 0 . Proposition If ( k a k k = 0 = S , then R n = S S n a n + 1 for each n N . Example D We approximate sin1 = ( k (2 k + 1)! k = 0 to three decimal places (error
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This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas at Austin.

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Calculus II Notes 12.5 - Calculus II-Stewart Dr. Berg...

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