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Connection between summation and integration

  1. Apr 18, 2014 #1
    If exist a connection between the infinitesimal derivative and the discrete derivative $$d = \log(\Delta + 1)$$ $$\Delta = \exp(d) - 1$$ exist too a coneection between summation ##\Sigma## and integration ##\int## ?
  2. jcsd
  3. Apr 18, 2014 #2
    Yep, both are special cases of measure-theoretic integration. Summation is with respect to the counting measure, regular integration is with respect to the Lebesgue measure.
  4. Apr 18, 2014 #3

    Stephen Tashi

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    Science Advisor

  5. Apr 18, 2014 #4
    Extremely complicated! I already know this article, but yet so I oponed this thread with the hope that could exist a simple formula and somebody that knew it could post it here...
  6. Apr 18, 2014 #5
    It's not complicated, it's just

    [tex]\int_n^m f(t) dt \sim \sum_{k=n}^m f(k) - \frac{f(m) + f(n)}{2}[/tex]

    If you want a better approximation, then it becomes more complicated.
  7. Apr 18, 2014 #6


    Staff: Mentor

    There's an old saying (due to Einstein, I believe), "Make things as simple as possible, but no simpler."
  8. Apr 18, 2014 #7
    "The simplicity is the most high degree of perfection."
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