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Thread: deducing a limit...

  1. #1
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    deducing a limit...

    Deduce that $\displaystyle \displaystyle\sum_{i=1}^n \frac{1}{n+i}\to log 2 $ as $\displaystyle n \to \infty $

    How on earth do i "deduce this"?! I've tried using $\displaystyle f(x) = \frac{1}{1 + x} $ but that didn't really get me anywhere...

    Thanks in advance!
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    It's a Riemann sum:

    $\displaystyle \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\frac{1}
    {{n + i}}} = \mathop {\lim }\limits_{n \to \infty } \frac{1}
    {n}\sum\limits_{i = 1}^n {\frac{1}
    {{1 + \dfrac{i}
    {n}}}} = \int_0^1 {\frac{1}
    {{1 + x}}\,dx} .$

    The rest follows.
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