Binomial Series - Calculus 2

In the last lesson we introduced two new types of power series known as Taylor series and Maclaurin series. Many functions can be represented as power series by forming Taylor series or Maclaurin series for those functions. However, there is a specific type of function that when represented as a Maclaurin series, results in a special type of Maclaurin series known as the Binomial series.

The Binomial series results from representing a function of the form (1+x)^k where that power of k can be any real number. The terms of the Binomial series are quite distinct and have a nice pattern that can be used to help represent functions of that form of (1+x)^k as a Maclaurin series. Functions like (1+x)^2, (1+x)^(1/2), (1+x)^-3, and (1+x)^(-3/2) are all examples of functions in this form that can be represented as a Maclaurin series by using the terms of the Binomial series. The interval of convergence for the resulting series is dependent on the value of k. We will look over the rules for determining the interval of convergence based on k in this lesson along with where the Binomial series comes from and how to use it. So with that, lets get right into this lesson on the Binomial series!

In this lesson, you will learn:

• The form of the Binomial series and its terms
• How to represent a function as Maclaurin series using the Binomial series
• How to determine the interval of convergence for the Binomial series

I hope you find this video to be helpful!

-Josh