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Thread: INequality with powers

  1. #1
    Senior Member I-Think's Avatar
    Apr 2009

    INequality with powers

    Need to solve

    $\displaystyle (\frac{2011}{n+1})^{n+1}<(\frac{2011}{n})^n
    Tried natural logs to get

    $\displaystyle log_e{ }2011<n(log_e(n+1)-log_e{ }n)+log_e(n+1)
    Stuck here

    Any ideas please?
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  2. #2
    Senior Member
    Nov 2010
    Staten Island, NY
    Perhaps cross multiply and cancel to get

    $\displaystyle 2011n^n<(n+1)^{n+1}$.

    This can be put in the form

    $\displaystyle 2011<(\frac{n+1}{n})^n(n+1)$.

    The right hand side is approximately $\displaystyle e(n+1)$ for large $\displaystyle n$.
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