In this final section of this chapter we are going to look
at another series representation for a function. Before we do this let’s first recall the
If n is any
positive integer then,
This is useful for expanding for large n
when straight forward multiplication wouldn’t be easy to do. Let’s take a quick look at an example.
Example 1 Use
the Binomial Theorem to expand
There really isn’t much to do other than plugging into the
Now, the Binomial Theorem required that n be a positive integer.
There is an extension to this however that allows for any number at all.
If k is any
number and then,
So, similar to the binomial theorem except that it’s an
infinite series and we must have in order to get convergence.
Let’s check out an example of this.
Example 2 Write
down the first four terms in the binomial series for
So, in this case and we’ll need to rewrite the term a little
to put it into the form required.
The first four terms in the binomial series is then,