Connection between summation and integration

[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 $\int$ ?

 

[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.

 

[Stephen Tashi]

Knowing jhenrique's tastes, we should also mention: http://en.wikipedia.org/wiki/Euler–Maclaurin_formula

 

[Jhenrique]

Knowing jhenrique's tastes, we should also mention: http://en.wikipedia.org/wiki/Euler–Maclaurin_formula

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...

 

[micromass]

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...

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.

 

[Mark44]

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

 

[Jhenrique]

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