chap11sec6

chap11sec6 - 11.6 Absolute Convergence and the Ratio and...

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11.6 Absolute Convergence and the Ratio and Root Tests Definition A series  n a  is called  absolutely convergent  if the series of absolute values  n a  is convergent. If the series  n a  has only positive terms then  | | 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  n a is absolutely convergent, then it is convergent. Example:  Determine whether the series is absolutely convergent, conditionally convergent or divergent.  
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chap11sec6 - 11.6 Absolute Convergence and the Ratio and...

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