PP 12.6 - Absolute Convergence and the Ratio and Root Tests...

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Absolute Convergence and the Ratio and Root Tests
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Definition A series Σ a n is called absolutely convergent if the series of absolute values Σ | a n | is convergent. n n = - 1 2 1 : Example = = = - 1 1 2 1 2 1 n n n n Both series are convergent geometric series by the Geometric Series Test. Hence, the original series is absolutely convergent.
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Example = - 1 ) 1 ( n n n We showed in a previous section that the alternating harmonic series is convergent. = = = - 1 1 1 ) 1 ( n n n n n We used the p-series test to show that the harmonic series is divergent. Hence, not all convergent series are absolutely convergent.
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Definition A series Σ a n is called conditionally convergent if it is convergent but not absolutely convergent. We just showed that the alternating harmonic series is conditionally convergent.
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Absolute Convergence Test If a series Σ a n is absolutely convergent, then it is convergent. Proof: n n n a a a 2 0 + convergent convergent absolutely n n a a convergent 2 n a ( 29 convergent + n n a a by the Comparison Test.
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We know that a sum of convergent series is convergent, so ( 29 . convergent is - + = n n n n a a a a Hence, series are conditionally convergent,
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This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.

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PP 12.6 - Absolute Convergence and the Ratio and Root Tests...

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