In general, summation refers to the addition of a sequence of any kind of number. The summation of infinite sequences is called a series, and involves the use of the concept of limits. While finite series can be expressed using addition, for longer series, or infinite series, writing out the entire series (e.g. 1 + 2 + 3 + 4 + 5...) is tedious. Instead, a method of denoting series, called sigma notation, can be used to efficiently represent the summation of many terms.

Sigma notation

Sigma notation is named based on its use of the capital Greek letter sigma:

When used in the context of mathematics, the capital sigma indicates that something (usually an expression) is being summed. For example, the sum of the integers 1, 2, 3, 4, and 5 can be expressed in sigma notation as:

The "n=1" is the lower bound of summation, and the 5 is the upper bound of summation, meaning that the index of summation starts out at 1 and stops when n equals 5. In the above example "n" is the expression. Therefore, to evaluate the summation above, start at n = 1 and evaluate the expression. Then move on to the next n, 2, and again evaluate the expression until the expression is evaluated for n = 5. The above example is a simple case since the index of summation and expression are equal. Expanding the above summation:

This is one of the simplest summation examples, just to illustrate sigma notation. Summations can get significantly more complicated. Below are a few more examples using slightly more complex expressions.


Evaluate the following summations.

1. :

2. :