Remember that there is an interval of convergence for each Taylor series, and within this interval of convergence, we can perform term-by-term integration and differentiation. So let us determine the interval of convergence for the Maclaurin series representation of e ^x. First of all, we will need to find the Maclaurin series for f (x) = e ^x.

f (x) = e ^x

f (0) = 1

f '(x) = e ^x

f '(0) = 1

f ''(x) = e ^x

f ''(0) = 1

Now let us determine the interval of convergence. If you need to review how to do this, then look at supplemental notes 26 for a quick review.

So this Maclaurin series converges for all x, and the interval of convergence is (-¥ ,¥ ).

For any Taylor or Maclaurin series, we can use the method we just used to determine the interval of convergence of a power series. The reason why we can do this is because Taylor and Maclaurin series are power series.

Here are some specific Maclaurin series and their interval of convergence that you should know.

There are more Maclaurin series stated in the textbook, but these are the ones that are used most often.

Suppose I want to find a Maclaurin series for f (x) = e ^{x ^ 2}. I could derive it out by finding a bunch of derivatives and then writing the series out, or I could use the fact that I know the Maclaurin series for e ^x, and then work from there. That is what I am going to do.

Now substitute x ² in for x.

Isn't that easy and quick?

EXAMPLE 1:

SOLUTION:

We know the following Maclaurin series

so

Now multiply all sides by x ².

EXAMPLE 2:

SOLUTION:

We know the following series.

We will do term-by-term differentiation to determine f (x).

Now take another derivative.

EXAMPLE 3:

Find the Maclaurin series for f (x) = sin x ².

SOLUTION:

We know the Maclaurin series for the sin x, so we will use it for a basis for the new series.

Sub in x ² for x.

Why do we use the Power series? We can use them in evaluating non-elementary integrals and limits of indeterminate forms. Here is an example of each.

EXAMPLE 4:

SOLUTION:

We do not know how to integrate this integral by hand, but we can integrate its related Maclaurin series.

The exact value using the fnint function on the calculator is 1.462651746. Our approximate value has an accuracy of two decimal places. If I would use more terms of this series, my approximation would become more accurate.

EXAMPLE 5:

SOLUTION:

 I know that it would have been easier to use l ' Hôpital's Rule, but it is a good example of using power series to approximate the value of a limit. You will learn in the coursework that you will take in engineering that sometimes power series are used to represent the solution of a certain differential equations. Therefore, it is good for you to see some of the applications of this type of series. Work through these examples taking note of the tricks that I used to define new Maclaurin series from given ones. Feel free to contact me if you have any questions on any of the topics discussed in this set of notes.