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 following theorem.
If n is any positive integer then,
This is useful for expanding (a+b)n 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 (2x−3)4
Solution. There really isn’t much to do other than plugging into the theorem.
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 |x|<1 then,
So, similar to the binomial theorem except that it’s an infinite series and we must have |x<1 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 √9−x
Solution. 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,