Sigma notation

From Graphic maths
Level: A Level Maths

Sigma notation is used to define the sum of a series, in a concise way.

Series of natural numbers

The sum of the first n natural numbers:

Can be written as:

This can be extended for powers of . For example the sum of the cubes of the first n natural numbers can be written like this:

Arithmetic progression

A general arithmetic progression takes the form:

The rth term is

The sum of the first terms can be written as:

We can simplify this equation slightly if we start the count at 0 rather than 1, ie if we count from , we only count up to . We can rewrite the sum like this:

Geometric progression

A general geometric progression takes the form:

The kth term is

The sum of the first terms can be written as:

We can simplify this equation slightly if we start the count at 0 rather than 1, ie if we count from , we only count up to . We can rewrite the sum like this:

General case

For function the sum

can be written as:

And the sum

can be written as:

See also