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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
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If n is any
positive integer then,
where,
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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.
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Example 1 Use
the Binomial Theorem to expand
Solution
There really isn’t much to do other than plugging into the
theorem.
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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
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If k is any
number and then,
where,
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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.
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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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