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.
Re-write 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:
$$
\cssId{s8}{\sum_{i=1}^{10}\ i^2}
\ \ \ \
\cssId{s9}{\text{ or }
\ \ \ \
\sum_{j=1}^{10}\ j^2}
\ \ \ \
\cssId{s10}{\text{ or }
\ \ \ \
\sum_{k=1}^{10}\ k^2}
\ \ \ \
\cssId{s11}{\text{ or }
\ \ \ \
\sum_{m=1}^{10}\ m^2}
\ \ \ \
\cssId{s12}{\text{ or }
\ \ \ \
\sum_{n=1}^{10}\ n^2}
$$
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:
$\displaystyle
\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\ }}}
\cssId{s38}{= -1 + 0 + 1 + 8}
\cssId{s39}{= 8}
$
$\displaystyle
\cssId{s40}{\sum_{k = 3}^7\ x_k}
\cssId{s41}{= x_3 + x_4 + x_5 + x_6 + x_7}
$
$\displaystyle
\cssId{s42}{\sum_{n = -2}^3\ 5}
\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\ \ }}}
\cssId{s44}{= 6\cdot 5}
\cssId{s45}{= 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:
$$
\begin{gather}
\cssId{s51}{\sum_{i=a}^b\ (x_i + y_i)}
\cssId{s52}{= \sum_{i=a}^b\ x_i
+
\sum_{i=a}^b\ y_i}\cr\cr
\cssId{s53}{\sum_{i=a}^b\ (x_i - y_i)}
\cssId{s54}{= \sum_{i=a}^b\ x_i
-
\sum_{i=a}^b\ y_i}
\end{gather}
$$
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:
$\displaystyle
\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}
$
$\displaystyle
\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
&\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:
$\displaystyle
\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:
$\displaystyle
\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.
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Mean, Median, and Mode