Sigma Notation for Series

A series is the sum of a sequence of numbers $a_n$ . For example

$1 + 4 + 9 + \cdots + n^2$

is the series of numbers $a_k = k^2$ for values of $k$ betweem $1$ and $n$ . Often, we are able to find an analytical expression for a series, which avoids the need to compute this large sum. For example, we can show that

$1 + 4 + 9 + \cdots + n^2 = \frac{n(n+1)(2n+1)}6$

We often use induction to demonstrate the proof of an analytical expression for a series. However, these results can become tedious to write if we have to keep track of all of the terms in these equations. Sigma notation is a much more compact way that we can write series, and keep track of the exact terms inside it. For example, we could represent the series $1 + 4 + 9 + \cdots + n^2$ is sigma notation as

$\sum_{i=1}^{n}{i^2}$

The symbol $\sum$ is the capitalised greek letter sigma, which makes the same sound as the letter 's' and so is used to represent a sum. The sigma notation contains two parts:

An index, represented by the $i = 1$ and $n$ below and above the $\sum$ . This tells us to start with the value $1$ , and keep incrementing it until we reach $n$ , which is the last term in our series.
An inner expression, which tells us what to do with the value $i$ in each term to get the value in the series.

The way to read the series $\sum_{i=1}^{n}{i^2}$ in sigma notation is:

For each value of $i$ , starting with $i = 1$ and ending with $i = n$ , square each value and take their sum.

If we follow this procedure, we would get the following

$i$	$i^2$
$1$	$1$
$2$	$4$
$3$	$9$
$\cdots$	$\cdots$
$n$	$n^2$

And so our series would be $1 + 4 + 9 + \cdots + n^2$ .

If necessary, we can also use brackets to apply summation to more complex inner expression, for example

$\sum_{i=1}^{n}{(i^2 -2i + 1)}$

In this case, we would evaluate $i^2 - 2i + 1$ for each value of $i$ , rather than just $i^2$

$i$	$i^2 -2i + 1$
$1$	$0$
$2$	$1$
$3$	$4$
$\cdots$	$\cdots$
$n$	$n^2 - 2n + 1$

...