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Math Help - Divergence of Series

  1. #1
    sportillo
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    Divergence of Series

    Can someone explain to me why one serie is convergent and the other one divergent? both seem to lead to 0 to me? Thx a lot for your help

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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by sportillo View Post
    Can someone explain to me why one serie is convergent and the other one divergent? both seem to lead to 0 to me? Thx a lot for your help

    http://www.math.uni-siegen.de/numeri...ine/img883.gif
    http://www.math.uni-siegen.de/numeri...ine/img888.gif
    The general term goes to zero as n becomes large, but in neither case
    can the sum be zero since they are the sums of positive terms and therefore
    the sum must be bigger than the first term.

    The first is the harmonic series, which can be shown to diverge by grouping
    the terms:

    <br />
\sum_1^{\infty} \frac{1}{k} = 1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4} \right)+\left( \frac{1}{5} + \frac{1}{6}+\frac{1}{7} + \frac{1}{8} \right)+ ...<br />

    Each group after the first has 2^{n-1} terms and the smallest term is 1/2^n and so each is greater 1/2 so:

    <br />
\sum_1^{\infty} \frac{1}{k} > 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2}+ ...<br />

    which shows that the series is divergent.

    Showing that the second converges is fairly simple, the integral test should do
    the job. Determing what it sums to is more tricky but Robin Chapman give a
    number of methods here.

    RonL
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