Sigma notation

Level: A Level Maths

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

Series of natural numbers

${\displaystyle 1+2+3+\cdots +n}$

Can be written as:

${\displaystyle \sum _{r\mathop {=} 1}^{n}r}$

This can be extended for powers of ${\displaystyle r}$. For example the sum of the cubes of the first n natural numbers can be written like this:

${\displaystyle \sum _{r\mathop {=} 1}^{n}r^{3}=1^{3}+2^{3}+3^{3}+\cdots +n^{3}}$

Arithmetic progression

A general arithmetic progression takes the form:

${\displaystyle a,a+d,a+2d,a+3d,\cdots }$

The rth term is

${\displaystyle a+(r-1)d}$

The sum of the first ${\displaystyle n}$ terms can be written as:

${\displaystyle \sum _{r\mathop {=} 1}^{n}{a+(r-1)d}}$

We can simplify this equation slightly if we start the count at 0 rather than 1, ie if we count from ${\displaystyle r=0}$, we only count up to ${\displaystyle r=n-1}$. We can rewrite the sum like this:

${\displaystyle \sum _{r\mathop {=} 0}^{n-1}{a+rd}}$

Geometric progression

A general geometric progression takes the form:

${\displaystyle a,ar,a{r^{2}},a{r^{3}},\cdots }$

The kth term is

${\displaystyle a{r^{k-1}}}$

The sum of the first ${\displaystyle n}$ terms can be written as:

${\displaystyle \sum _{k\mathop {=} 1}^{n}a{r^{k-1}}}$

We can simplify this equation slightly if we start the count at 0 rather than 1, ie if we count from ${\displaystyle r=0}$, we only count up to ${\displaystyle r=n-1}$. We can rewrite the sum like this:

${\displaystyle \sum _{k\mathop {=} 0}^{n-1}a{r^{k}}}$

General case

For function ${\displaystyle F(r)}$ the sum

${\displaystyle F(1)+F(2)+F(3)\cdots +F(n)}$

can be written as:

${\displaystyle \sum _{r\mathop {=} 1}^{n}F(r)}$

And the sum

${\displaystyle F(m)+F(m+1)+F(m+2)\cdots +F(n-1)+F(n)}$

can be written as:

${\displaystyle \sum _{r\mathop {=} m}^{n}F(r)}$