Sigma Notation and Series

$\sum_{r=1}^{n} f(r) = f(1) + f(2) + ... + f(n)$

Standard Results

$\sum_{r=1}^{n} 1 = n \qquad \sum_{r=1}^{n} r = \frac{n(n+1)}{2} \qquad \sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6} \qquad \sum_{r=1}^{n} r^3 = \left[\frac{n(n+1)}{2}\right]^2$

Properties

$\sum(af(r) + bg(r)) = a\sum f(r) + b\sum g(r)$

$\sum_{r=1}^{10} (3r+1) = 3\sum r + \sum 1 = 3(55) + 10 = 175$.

$\sum_{r=1}^{n} (2r^2 - r) = \frac{n(n+1)(2n+1)}{3} - \frac{n(n+1)}{2}$.