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Math Help - Euler–Mascheroni constant

  1. #1
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    Euler–Mascheroni constant

    I have this doubt regarding the Euler–Mascheroni constant γ . It is integral 1/[x] - 1/x limits from 1 to infinity and it is value is
    0.57721 . Of cousre the big question that remains is if this is rational or not .
    Let us say i keep splitting the limits i.e from 1 to 2 then 2 to 3 and so on from n to n+1 . I use limit n -> infinity.
    The value would be like this 1 - (log 2 - log1 ) + 1/2 - ( log 3 - log 2) + ... 1/n - ( log (n+1) - log n ) = γ.
    Firstly sigma 1/n should have been approx to log ( n+1) rather than log(n) .
    In such a case the error would be .5 .
    So why do we approx to me log(n) and what is the specialty of the constant .
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  2. #2
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    Quote Originally Posted by kaushiks.nitt View Post
    I have this doubt regarding the Euler–Mascheroni constant γ . It is integral 1/[x] - 1/x limits from 1 to infinity and it is value is
    0.57721 . Of cousre the big question that remains is if this is rational or not .
    Let us say i keep splitting the limits i.e from 1 to 2 then 2 to 3 and so on from n to n+1 . I use limit n -> infinity.
    The value would be like this 1 - (log 2 - log1 ) + 1/2 - ( log 3 - log 2) + ... 1/n - ( log (n+1) - log n ) = γ.
    Firstly sigma 1/n should have been approx to log ( n+1) rather than log(n) .
    In such a case the error would be .5 .
    So why do we approx to me log(n) and what is the specialty of the constant .
    Read:

    Euler-Mascheroni Constant -- from Wolfram MathWorld

    Euler?Mascheroni constant - Wikipedia, the free encyclopedia

    Applications of the Euler–Mascheroni constant to physics are numerous eg. Bessel's and Helmholtz' equations (heat conduction, electromagnetic waves, etc.), regularization of Feynman diagrams in QFT etc. etc.
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  3. #3
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    I already read about y in both the links you have mentioned.
    I know even how it is derived .
    My question is there is an obvious miscalculation and
    log(n+1) is a better approx to sigma(1/n) than log(n).
    So why do we don't use this ??
    If you have a solution to this question .
    Reply or else it is ok.
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  4. #4
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    There are two things you should never try to prove ...... the impossible and the obvious.

    This is funny. Even if it is obvious you will have to check it and it is always better to have your own proof.
    And what do u consider impossible .
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