﻿ Summation Notation

# Summation Notation

Summation notation gives a compact way to represent sums, when the terms exhibit some common pattern.

For example, consider this sum:

$$\cssId{s3}{1+4+9+16+25+36+49+64+81+100}$$

Each term is a perfect square. Rewrite the sum to clearly show the pattern:

$$\cssId{s6}{1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2}$$

Using summation notation, the five most common representations for this sum are:

$$\begin{gather} \cssId{s8}{\sum_{i=1}^{10}\ i^2} \ \ \ \ \cssId{s9}{\text{ or } \ \ \ \ \sum_{j=1}^{10}\ j^2} \ \ \ \ \cssId{s10}{\text{ or } \ \ \ \ \sum_{k=1}^{10}\ k^2}\cr\cr \cssId{s11}{\text{ or } \ \ \ \ \sum_{m=1}^{10}\ m^2} \ \ \ \ \cssId{s12}{\text{ or } \ \ \ \ \sum_{n=1}^{10}\ n^2} \end{gather}$$

Summation notation is also called sigma notation.

Using $\sum_{i=1}^{10}\ i^2\,$ as an example:

• $\sum_{i=1}^{10}\ i^2\,$ is an expression; it's a number. Like all numbers, it has lots of different names.
• The expression looks like  $\,\sum_{i=1}^{10}\ i^2\,$  when it appears in a line of text; this is a more vertically-compact version. When it doesn't need to be vertically constrained, then it looks like this:   $\displaystyle\sum_{i=1}^{10}\ i^2\,$
• $\sum_{i=1}^{10}\ i^2\,$ is read aloud as:  ‘the sum, as $\,i\,$ goes from $\,1\,$ to $\,10\,,$ of $\,i\,$ squared’
• The variable $\,i\,$ is called the index of summation.

The five most common letters to use for the index of summation are $\,i\,,$ $\,j\,,$ $\,k\,,$ $\,m\,$ and $\,n\,.$

• The number $\,1\,$ in ‘$\,i=1\,$’ is called the lower limit of summation. This gives the starting value for $\,i\,.$
• The number $\,10\,$ is the upper limit of summation. This gives the ending value for $\,i\,.$

$\,i\,$ starts with the lower limit and is incremented by one until the upper limit is reached.

• The expression next to the summation symbol gives the pattern for each of the terms in the sum.

## Examples

\begin{align} \cssId{s36}{\sum_{j = -1}^2\ j^3}\ &\cssId{s37}{= \overset{j=-1}{\overbrace{(-1)^3}} + \overset{j=0}{\overbrace{\ 0^3\ }} +\overset{j=1}{\overbrace{\ 1^3\ }} + \overset{j=2}{\overbrace{\ 2^3\ }}}\cr &\cssId{s38}{= -1 + 0 + 1 + 8}\cr\cr &\cssId{s39}{= 8} \end{align}
$$\cssId{s40}{\sum_{k = 3}^7\ x_k} \cssId{s41}{= x_3 + x_4 + x_5 + x_6 + x_7}$$
\begin{align} &\cssId{s42}{\sum_{n = -2}^3\ 5}\cr &\qquad \cssId{s43}{= \overset{n=-2}{\overbrace{\ \ 5\ \ }} + \overset{n=-1}{\overbrace{\ \ 5\ \ }} + \overset{n=0}{\overbrace{\ \ 5\ \ }} + \overset{n=1}{\overbrace{\ \ 5\ \ }} + \overset{n=2}{\overbrace{\ \ 5\ \ }} + \overset{n=3}{\overbrace{\ \ 5\ \ }}}\cr\cr &\qquad \cssId{s44}{= 6\cdot 5} \cssId{s45}{= 30} \end{align}
PROPERTIES OF SUMS

Let $\,a\,$ and $\,b\,$ be integers with $\,a\lt b\,,$ and let $\,k\,$ be any real number.

You can split sums and differences apart:

$$\begin{gather} \cssId{s51}{\sum_{i=a}^b\ (x_i + y_i)} \cssId{s52}{= \sum_{i=a}^b\ x_i + \sum_{i=a}^b\ y_i}\cr\cr \cssId{s53}{\sum_{i=a}^b\ (x_i - y_i)} \cssId{s54}{= \sum_{i=a}^b\ x_i - \sum_{i=a}^b\ y_i} \end{gather}$$

You can slide a constant out of a sum:

$$\cssId{s56}{\sum_{i=a}^b\ kx_i} \cssId{s57}{= k\ \sum_{i=a}^b\ x_i }$$

To sum a constant value, you must correctly count the number of terms:

$$\cssId{s59}{\sum_{i=a}^b\ k} \cssId{s60}{= (b-a+1)\cdot k}$$

## Why Do These Properties Work?

The following examples illustrate why these properties work. They are all easy consequences of the commutative (re-ordering) and associative (re-grouping) properties of addition, and the distributive law:

### Splitting Sums and Differences Apart

\begin{align} \cssId{s65}{\sum_{i=1}^2 \ (x_i + y_i)}\ &\cssId{s66}{= (x_1 + y_1) + (x_2 + y_2)}\cr &\cssId{s67}{= (x_1 + x_2) + (y_1 + y_2)} \cr &\cssId{s68}{= \sum_{i=1}^2\ x_i + \sum_{i=1}^2\ y_i} \end{align}

\begin{align} \cssId{s69}{\sum_{i=1}^2 \ (x_i - y_i)}\ &\cssId{s70}{= (x_1 - y_1) + (x_2 - y_2)}\cr &\cssId{s71}{= (x_1 + x_2) - y_1 - y_2}\cr\cr &\cssId{s72}{= (x_1 + x_2) - (y_1 + y_2)}\cr &\cssId{s73}{= \sum_{i=1}^2\ x_i - \sum_{i=1}^2\ y_i} \end{align}

### Sliding a Constant out of a Sum

\begin{align} \cssId{s75}{\sum_{i=1}^2\ kx_i}\ &\cssId{s76}{= kx_1 + kx_2}\cr &\cssId{s77}{= k(x_1 + x_2)}\cr &\cssId{s78}{= k\ \sum_{i=1}^2\ x_i} \end{align}

### Summing a Constant Value

\begin{align} \cssId{s80}{\sum_{i=3}^7\ k}\ \ &\cssId{s81}{= \overset{i=3}{\overbrace{\ \ k\ \ }} + \overset{i=4}{\overbrace{\ \ k\ \ }} + \overset{i=5}{\overbrace{\ \ k\ \ }} +\overset{i=6}{\overbrace{\ \ k\ \ }} + \overset{i=7}{\overbrace{\ \ k\ \ }}} \cr &\cssId{s82}{= \underset{\text{upper limit - lower limit + 1}} {\underbrace{ \overset{\text{How many terms?}}{\overbrace{(7-3+1)}} }}} \cssId{s83}{\cdot\ \ k} \end{align}

Notice that you can't count the terms by just taking the upper limit and subtracting the lower limit; you must add one to this difference.