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Math Help - logarithm function

  1. #1
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    logarithm function

    Let n be a positive integer greater than 2.

    a)Find the greatest integer k for which
    1/2 + 1/3 + ... + 1/k < ln(n)

    b)Find the least integer k for which
    ln(n) < 1 + 1/2 + 1/3 + ... + 1/k


    Thanks for the help.
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  2. #2
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    Quote Originally Posted by elmo View Post
    Let n be a positive integer greater than 2.
    a)Find the greatest integer k for which
    1/2 + 1/3 + ... + 1/k < ln(n)

    b)Find the least integer k for which
    ln(n) < 1 + 1/2 + 1/3 + ... + 1/k
    This is Napierís inequality: 0 < a < b\; \Rightarrow \;\frac{1}{b} \leqslant \frac{{\ln (b) - \ln (a)}}{{b - a}} \leqslant \frac{1}{a}.
    From which it follows that: \frac{1}{{N + 1}} \leqslant \ln (N + 1) - \ln (N) \leqslant \frac{1}{N}.

    So we get \sum\limits_{k = 1}^{N - 1} {\frac{1}{{k + 1}}}  \leqslant \ln (N) \leqslant \sum\limits_{k = 1}^{N - 1} {\frac{1}{k}} .
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