Summation Nesting

Double Summation

Example of double summations arise in the analysis of nested loops in computed programs.

Example

${\sum }_{i=1}^4{\sum }_{j=1}^{3}ij$

To evaluate the double sum, first expand the inner summation and then the outer summation.

Example

$$\begin{array}{rl}& \sum _{i=1}^4\sum _{j=1}^{3}ij=\sum _{i=1}^4(i+2i+3i)\\ & =\sum _{i=1}^{4}6i\\ & =6+12+18+24=60\end{array}$$