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Math Help - Alternating series convergence?

  1. #1
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    Alternating series convergence?

    Why is sigma (-1)^n/n a convergent series while sigma 1/n is a divergent p-series?


    ...is it because half the terms are used for either sign +/- and therefore each set decays more rapidly than 1/n?
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  2. #2
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    Let a_n=\frac1n, then obviously \lim_{n\to\infty}\frac1n=0. On the other hand, a_n is a strictly decreasing sequence for n\ge1. Hence, by Leibniz test the series converges.

    But, this series converges conditionally, since \left| a_{n} \right|=\frac{1}{n} and \sum\limits_{n=1}^{\infty }{\frac{1}{n}} diverges.
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