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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,

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