Summation

General rules

$\displaystyle \sum^n_{r=1}c = c\cdot n$

$\displaystyle \sum^n_{r=1}c\cdot a_r = c\cdot \sum^n_{r=1} a_r$

$\displaystyle \sum^n_{r=1}a_r + b_r = \sum^n_{r=1} a_r + \sum^n_{r=1} b_r$

Rules

$\displaystyle \sum^n_{r=1}r = \frac{n(n+1)}{2}$

$\displaystyle \sum^n_{r=1}r^2 = \frac{n(n+1)(2n+1)}{6}$

$\displaystyle \sum^n_{r=1}r^3 = \frac{n^2(n+1)^2}{4}$

Riemann sums

\[\int^b_af(x)dx = \lim_{n\to\infty}\sum^n_{r=1}f(x_r)\Delta x\] \[\textstyle x_r = a+\frac{r}{n}(b-a)\] \[\textstyle \Delta x=\frac{b-a}{n}\]

Where $n$ is the number of divisions which determines the accuracy.