In this section we need to do a brief review of summation
notation or sigma notation. We’ll start
out with two integers, n and m, with and a list of numbers denoted as follows,
We want to add them up, in other words we want,
For large lists this can be a fairly cumbersome notation so
we introduce summation notation to denote these kinds of sums. The case above is denoted as follows.
The i is called
the index of summation. This notation
tells us to add all the a_{i}’s
up for all integers starting at n and
ending at m.
For instance,
Properties
Here are a couple of formulas for summation notation.
Note that we started the series at to denote the fact that they can start at any
value of i that we need them to. Also note that while we can break up sums and
differences as we did in 2 above we
can’t do the same thing for products and quotients. In other words,
Formulas
Here are a couple of nice formulas that we will find useful
in a couple of sections. Note that these
formulas are only true if starting at . You can, of course, derive other formulas
from these for different starting points if you need to.
Here is a quick example on how to use these properties to
quickly evaluate a sum that would not be easy to do by hand.
Example 1 Using
the formulas and properties from above determine the value of the following
summation.
Solution
The first thing that we need to do is square out the stuff
being summed and then break up the summation using the properties as follows,
Now, using the formulas, this is easy to compute,
Doing this by hand would definitely taken some time and
there’s a good chance that we might have made a minor mistake somewhere along
the line.
