Binomial Series

# Binomial Series Notes | Study Algebra for IIT JAM Mathematics - Mathematics

## Document Description: Binomial Series for Mathematics 2022 is part of Algebra for IIT JAM Mathematics preparation. The notes and questions for Binomial Series have been prepared according to the Mathematics exam syllabus. Information about Binomial Series covers topics like and Binomial Series Example, for Mathematics 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Binomial Series.

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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,

The document Binomial Series Notes | Study Algebra for IIT JAM Mathematics - Mathematics is a part of the Mathematics Course Algebra for IIT JAM Mathematics.
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