﻿ Summation Notation
SUMMATION NOTATION
• PRACTICE (online exercises and printable worksheets)

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

For example, consider this sum: $$1+4+9+16+25+36+49+64+81+100$$ Each term is a perfect square.
Re-write the sum to clearly show the pattern: $$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2$$ Using summation notation, the five most common representations for this sum are: $$\sum_{i=1}^{10}\ i^2 \ \ \ \ \text{ or } \ \ \ \ \sum_{j=1}^{10}\ j^2 \ \ \ \ \text{ or } \ \ \ \ \sum_{k=1}^{10}\ k^2 \ \ \ \ \text{ or } \ \ \ \ \sum_{m=1}^{10}\ m^2 \ \ \ \ \text{ or } \ \ \ \ \sum_{n=1}^{10}\ n^2$$ 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:
$\displaystyle \sum_{j = -1}^2\ j^3 = \overset{j=-1}{\overbrace{(-1)^3}} + \overset{j=0}{\overbrace{\ 0^3\ }} +\overset{j=1}{\overbrace{\ 1^3\ }} + \overset{j=2}{\overbrace{\ 2^3\ }} = -1 + 0 + 1 + 8 = 8$
$\displaystyle \sum_{k = 3}^7\ x_k = x_3 + x_4 + x_5 + x_6 + x_7$
$\displaystyle \sum_{n = -2}^3\ 5 = \overset{n=-2}{\overbrace{\ \ 5\ \ }} + \overset{n=-1}{\overbrace{\ \ 5\ \ }} + \overset{n=0}{\overbrace{\ \ 5\ \ }} + \overset{n=1}{\overbrace{\ \ 5\ \ }} + \overset{n=2}{\overbrace{\ \ 5\ \ }} + \overset{n=3}{\overbrace{\ \ 5\ \ }} = 6\cdot 5 = 30$
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: $$\begin{gather} \sum_{i=a}^b\ (x_i + y_i) = \sum_{i=a}^b\ x_i + \sum_{i=a}^b\ y_i\cr\cr \sum_{i=a}^b\ (x_i - y_i) = \sum_{i=a}^b\ x_i - \sum_{i=a}^b\ y_i \end{gather}$$
You can slide a constant out of a sum:
$$\sum_{i=a}^b\ kx_i = k\ \sum_{i=a}^b\ x_i$$ To sum a constant value, you must correctly count the number of terms: $$\sum_{i=a}^b\ k = (b-a+1)\cdot k$$
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:

\displaystyle \begin{align} \sum_{i=1}^2 \ (x_i + y_i) &= (x_1 + y_1) + (x_2 + y_2)\cr &= (x_1 + x_2) + (y_1 + y_2) \cr &= \sum_{i=1}^2\ x_i + \sum_{i=1}^2\ y_i \end{align}

\displaystyle \begin{align} \sum_{i=1}^2 \ (x_i - y_i) &= (x_1 - y_1) + (x_2 - y_2)\cr &= (x_1 + x_2) - y_1 - y_2 \cr &= (x_1 + x_2) - (y_1 + y_2)\cr &= \sum_{i=1}^2\ x_i - \sum_{i=1}^2\ y_i \end{align}
• Sliding a constant out of a sum:

\displaystyle \begin{align} \sum_{i=1}^2\ kx_i &= kx_1 + kx_2\cr &= k(x_1 + x_2)\cr &= k\ \sum_{i=1}^2\ x_i \end{align}
• Summing a constant value:

\displaystyle \begin{align} \sum_{i=3}^7\ k\ &= \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 &= \underset{\text{upper limit - lower limit + 1}} {\underbrace{ \overset{\text{How many terms?}}{\overbrace{(7-3+1)}} }} \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.

Master the ideas from this section