Taylor Series & Maclaurin Series - Calculus 2

In the past few lessons we have focused on power series, and how we can represent functions as a power series. Specifically, we were representing functions similar to the form of the sum of a geometric series as a power series. However, this is not the only way to represent a function as a power series. Another way to represent a function as a power series is to represent it as one of two special types of power series known as the Taylor series and Maclaurin series.

The Taylor series is derived from analyzing the derivatives of a power series, and generalizing them with an nth derivative. This nth derivative leads to the form of a Taylor series and subsequently a Maclaurin series. Taylor series are still power series that are centered at some value c, and Maclaurin series are just Taylor series where that value of c is 0. To represent a function as a Taylor series or Maclaurin series, we will need to find the nth derivative of that function evaluated at the value of c, and then substitute that into the form for each series respectively.

Some Maclaurin series are referred to as "common" Maclaurin series, as they are series that represent functions that are commonly used. This includes sin(x), cos(x), ln(x+1), e^x, and more! We can use these common Maclaurin series to help represent similar functions as Maclaurin series without needing to start from scratch and find the nth derivative evaluated at c for that function. Representing functions like sin(2x), x*cos(5x), or e^(2x) are just a few examples of functions where this method is useful. We will take a look at this method as well as everything else mentioned above in this new lesson, so let's get to it!

In this lesson, you will learn:

• The form of Taylor series & Maclaurin series
• How to represent a function as a Taylor series or Maclaurin series
• How to use common Maclaurin series to represent functions as Maclaurin series

I hope you find this video to be helpful!

-Josh