Did you know that there is a direct connection between Taylor Series and the Binomial Expansion?
Yep, the Binomial Series is a special case of the Maclaurin series (Taylor Series centered at zero) or Power Series, and it occurs so often that it’s definitely an expansion formula that you want to commit to memory!
Okay, so before we jump into the Binomial Series, we have to take a step back and talk about the Binomial Theorem or Binomial Expansion.
Now as we know, the Binomial Theorem is a way of multiplying out a binomial expression that is raised to some large power of n, where n is some positive integer and is the exponent on the binomial expression.
If you don’t remember the Binomial Theorem, or you’re feeling a little rusty with the concept, I encourage you to look back at that video because the Binomial Series and the Binomial Theorem go hand-in-hand.
The Binomial Series Explained
So, the Binomial Theorem enables us to quickly write a binomial in expanded form, but it has a limitation. We can only use it for when our exponent is a natural number.
Thankfully, one of Sir Isaac Newton’s great accomplishments was to extend the Binomial Theorem so that our exponent is any real number (i.e., k is positive or negative), as nicely stated by Paul’s Online Notes.
Together we are going to learn how to use the Binomial Series to expand a function as a Power Series for four or five terms using easy to follow steps.
The process is super easy and straightforward, as we will see after looking at several examples together, and will be a very helpful tool in your tool-belt!
Binomial Series – Video
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