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Calculus II - Notes
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## Binomial Series

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  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     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  then,                                 where,

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      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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