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 || nn aa = We have seen that the alternating harmonic series 1 1 (1 ) n n n = is convergent. However, it is not absolutely convergent because 1 11 ) 1 n ∞∞ == = ∑∑ is the harmonic series and is divergent. So we would say that 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 ) 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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This note was uploaded on 01/20/2012 for the course MATH 2057 taught by Professor Estrada during the Fall '08 term at LSU.

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

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