# Riemann Sum

1. Sep 14, 2008

### Brunll

$$\frac{n(n+1)}{2}, \frac{n(n+1)(2n+1)}{6} ,(\frac{n(n+1)}{2})^2$$

How were they evolved?

And what about $$i^4, i^5, i^6,...$$

Thankyou everybody;;

2. Sep 14, 2008

### HallsofIvy

Staff Emeritus
Those are, of course, $\sum_{i=1}^n i$, $\sum_{i=1}^n i^2$, and $\sum_{i=1}^n i^3$, respectively. You can sum such things by using "Newton's Divided Difference" formula. It is a discrete version of Taylor's series. If f(n) is a function defined on the non-negative integers, $\Delta f$ is the "first difference", f(n)- f(n-1), $\Delta^2 f$ is the "second difference", $\Delta f(n)- \Delta f(n-1)$, $\Delta^3 f$ is the "third difference", etc. then