Examples of summation expressed in Sigma notation

Consider the sigma notation:

\[\sum_{r=1}^{14} r^2.\]

Unpacking this notation in its expanded form:

\[\sum_{r=1}^{14} r^2 = 1^2 + 2^2 + 3^2 + \dots + 14^2.\]

Unpacking the following sigma notation

\[\sum_{r=0}^{6} \frac{r}{(r+1)(r+5)}\]

gives

\[\frac{0}{(1)(5)} + \frac{1}{(2)(6)} + \frac{2}{(3)(7)} + \frac{3}{(4)(8)} + \dots + \frac{6}{(7)(11)}.\]