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Calculus II Notes 12.6

# Calculus II Notes 12.6 - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 12.6 12.6 Absolute Convergence; and Ratio and Root Tests Theorem Absolute Convergence If a k converges, then a k converges. Proof: For each k N we have a k a k a k so 0 a k + a k 2 a k . If a k converges, then 2 a k converges. By the basic comparison theorem, a k + a k ( ) converges, so a k = a k + a k ( ) a k converges. Definition A series a k is absolutely convergent if the series of absolute values a k is convergent. Examples A 1 ( ) k k 2 converges absolutely since 1 ( ) k k 2 = 1 k 2 is a convergent p-series. Example B 1 ( ) k k + 1 k = 0 converges (to ln2 ), but is not absolutely convergent since 1 ( ) k k + 1 k = 0 = 1 k + 1 k = 0 is divergent. Definition A series a k is conditionally convergent if it converges, but the series of absolute values a k diverges. Note: From example B, 1 ( ) k k + 1 k = 0 is conditionally convergent.

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Calculus II Notes 12.6 - Calculus II-Stewart Dr Berg Spring...

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