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Math Help - Show that this series diverges

  1. #1
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    Show that this series diverges

    Hi, how do I show that this series is divergent?

    \sum_{n=1}^{\infty} ln \left( \frac{n+1}{n} \right)

    Thanks.
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  2. #2
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    Quote Originally Posted by coldfire View Post
    Hi, how do I show that this series is divergent?

    \sum_{n=1}^{\infty} ln \left( \frac{n+1}{n} \right)

    Thanks.
    I suggest using a ratio test and then a series of log rules to simplify the expression.
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  3. #3
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    Quote Originally Posted by coldfire View Post
    Hi, how do I show that this series is divergent?

    \sum_{n=1}^{\infty} ln \left( \frac{n+1}{n} \right)

    Thanks.
    Note that

    \ln \left( \frac {n+1}{n}\right) = \ln (n+1) - \ln n

    so the series is telescopic.
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  4. #4
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    Hello, coldfire

    Show that this series is divergent:

    . . \sum_{n=1}^{\infty}\ln\left(\frac{n+1}{n} \right)

    We have: . \sum^{\infty}_{n=1}\ln\left(\frac{n+1}{n}\right)

    . . . . . . =\;\;\lim_{n\to\infty}\bigg[\ln\left(\frac{2}{1}\right) + \ln\left(\frac{3}{2}\right) + \ln\left(\frac{4}{3}\right) + \ln\left(\frac{5}{4}\right) + \hdots + \ln\left(\frac{n+1}{n}\right)\bigg]

    . . . . . . =\;\;\lim_{n\to\infty}\bigg[\ln\left(\frac{\rlap{/}2}{1}\cdot\frac{\rlap{/}3}{\rlap{/}2}\cdot\frac{\rlap{/}4}{\rlap{/}3}\cdot\frac{\rlap{/}5}{\rlap{/}4}\:\cdots\:\frac{n+1}{\rlap{/}n}\right)\bigg]


    . . . . . . =\;\;\lim_{n\to\infty}\ln\,(n+1) \;\;=\;\;\ln(\infty) \;\;=\;\;\infty

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