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.

Comments on Summation 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.