# 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

Staff Emeritus
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

Staff Emeritus
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."