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ch11s6

# ch11s6 - 11.6 Absolute Convergence and the Ratio and Root...

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11.6 Absolute Convergence and the Ratio and Root Tests Definition A series is called absolutely convergent if the series of absolute values n a n a is convergent. If the series has only positive terms then . n a | | n n a a = We have seen that the alternating harmonic series 1 1 ( 1) n n n = is convergent. However, it is not absolutely convergent because 1 1 1 ( 1) 1 n n n n n = = = is the harmonic series and is divergent. So we would say that 1 1 ( 1) n n n = is conditionally convergent because it is convergent but not absolutely convergent. n a is conditionally convergent if it is convergent but not absolutely convergent. Theorem: If a series is absolutely convergent, then it is convergent. n a Example: Determine whether the series is absolutely convergent, conditionally convergent or divergent. 1 ( 1) n n n = . Note: First check for absolute convergence since that would mean it is convergent. If it is not absolutely convergence then check for conditional convergence.

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ch11s6 - 11.6 Absolute Convergence and the Ratio and Root...

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