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Math Help - Limit

  1. #1
    Junior Member
    Joined
    Jan 2009
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    Limit

    Hi, can someone evaluate this limit? I need it for a mathematical model, so, technically, I'm allowed to use technology (Mathematica gives the result as approximately 23.7), but if someone knows an easy way to solve this, it would be better.

    \lim_{n \to \infty} \frac{90n}{\displaystyle\sum_{i=0}^{4n} 0.975^ \frac{i}{n}}=?

    Thanks.
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  2. #2
    Member
    Joined
    May 2009
    Posts
    127
    I get the limit to be:

    -90 \frac{\ln 0.975}{1-0.975^4} = 23.6585 \dots

    The way I did this was to notice that the denominator is a geometric progression where r = 0.975^{1/n} and so use the formula for the sum of a geometric progression to re-express the denominator.

    Then, approaching the limit, the behaviour of the denominator is easy and the numerators' can be found by expanding in terms of a power series.

    It's as easy as that!
    Last edited by the_doc; May 16th 2009 at 03:59 PM. Reason: Corrected a missing negative sign!
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