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# PP 12.5_1 - Alternating Series These are series whose terms...

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Alternating Series These are series whose terms are alternately positive and negative. n n n a = - 1 ) 1 ( n n n a = - - 1 1 ) 1 ( n n n a = + - 1 1 ) 1 (

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Comments The p-series test and the Comparison Tests only apply to series with positive terms. The Geometric Series Test can be used for series with positive terms as well as for alternating series since | r | controls the behavior of the series. The Test for Divergence applies to all types or series, but has limited use.
Example e. convergenc for 3 2 ) 1 ( Test 1 n n n = - This is clearly an alternating series since the powers of –1 become 1 or –1 as n runs over the integers. 1 1 1 1 3 2 3 2 3 2 3 2 ) 1 ( - = = = - - = - = - n n n n n n n So, we have a geometric series with | r = -2/3| < 1 . Hence the series is convergent by the Geometric Series Test.

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Comments Most alternating series are NOT geometric series.
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PP 12.5_1 - Alternating Series These are series whose terms...

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