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Math Help - Convergence

  1. #1
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    Convergence

    Let \{a_n\} be a strictly increasing sequence of positive integers.
    Can the series
    \sum^{\infty}_{n=1}\left(1-\frac{a_n}{a_{n+1}}\right)=\left(1-\frac{a_1}{a_2}\right)+\left(1-\frac{a_2}{a_3}\right)+\ldots
    ever converge?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by math sucks View Post
    Let \{a_n\} be a strictly increasing sequence of positive integers.
    Can the series
    \sum^{\infty}_{n=1}\left(1-\frac{a_n}{a_{n+1}}\right)=\left(1-\frac{a_1}{a_2}\right)+\left(1-\frac{a_2}{a_3}\right)+\ldots
    ever converge?
    At first glance it would appear to be bounded below by the harmonic series (if I've got the right name for it)
    \sum_{n = 1}^{\infty}\frac{1}{n}
    which is divergent.

    -Dan
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  3. #3
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    Quote Originally Posted by topsquark View Post
    At first glance it would appear to be bounded below by the harmonic series (if I've got the right name for it)
    \sum_{n = 1}^{\infty}\frac{1}{n}
    which is divergent.

    -Dan
    Can you prove your statement?
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