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Math Help - Absolute convergence problem

  1. #1
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    Absolute convergence problem

    how would you determine if if the series converges absolutely, conditionally or diverges?

    series n=1 to infinity (-5)^-n
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  2. #2
    o_O
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    Let a_n = (-5)^{-n} = (-1)^{-n}5^{-n}. Then |a_n| = 5^{-n}.

    Now, notice that: \sum |a_n| = \sum 5^{-n} = \sum \left(\frac{1}{5}\right)^n

    which is a geometric series.
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  3. #3
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    Quote Originally Posted by o_O View Post
    Let a_n = (-5)^{-n} = (-1)^{-n}5^{-n}. Then |a_n| = 5^{-n}.

    Now, notice that: \sum |a_n| = \sum 5^{-n} = \sum \left(\frac{1}{5}\right)^n

    which is a geometric series.
    Just to add \sum_{n=1}^\infty (-5)^{-n} = \sum_{n=1}^\infty \left(\frac{-1}{5}\right)^n is also geometric.
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