Connection between summation and integration

1. Apr 18, 2014

Jhenrique

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. Apr 18, 2014

micromass

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.

3. Apr 18, 2014

Stephen Tashi

4. Apr 18, 2014

Jhenrique

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...

5. Apr 18, 2014

micromass

It's not complicated, it's just

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

If you want a better approximation, then it becomes more complicated.

6. Apr 18, 2014

Staff: Mentor

There's an old saying (due to Einstein, I believe), "Make things as simple as possible, but no simpler."

7. Apr 18, 2014

Jhenrique

"The simplicity is the most high degree of perfection."