Binomial Series | Algebra - Mathematics PDF Download

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.

Binomial Theorem
If n is any positive integer then,

where, 

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.

Binomial Series

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,