Summation Operator

Summation Operator Annotated

The summation operator is represented by the symbol (summation) and represents the operation of summing a sequence of expressions together. The operator is typically used in an expression like this:

In plain language, this means sum the sequence of numbers together represented by the expression starting from and iterating until . At each step, the element of the sequence is calculated by substituting the current value of into the expression. The sequence of expressions is shown below.

Expression Variable

The expression on top of the summation operator, in this case , controls the length of the sequence being added together. Expanding the expressions as a summation gives the following.

Adding together the sequence of expressions gives the result.


The summation operator is used extensively in calculus, probability and other areas of math to express summations. For example, it is used extensively to define sequences and series.


Calculate Sum of Counting Numbers

To calculate the summation of the counting numbers 1 + 2 + 3 + ... + n the sequence can be visualized geometrically and solved for by finding the area of the shape formed.