New Series from Old

Purpose

The purpose of this lab is to introduce you to some useful techniques for generating Taylor series, including substitution, multiplication, and term-by-term integration and differentiation.

Background

The general form of the Taylor series representation with base point x=a of a function is given by the following

In class we have derived the following four important Taylor series.

where the last series is really the formula for the sum of a geometric series in disguise.

Once you have a Taylor series representation for a function, it can be used in several ways to generate Taylor series representations of related functions. Here are some examples of four useful techniques.

Substitution

To get a Taylor series for , you could go through the standard procedure of differentiating and substituting into the general formula, but an easier (and also correct) procedure is to take the series for and substitute 2x for x, obtaining

Writing out the first few terms gives

Here is how you could compare the two approaches using Maple for this example and one of the examples shown below.

  > with(CalcP):

  > Taylor(exp(2*x),x=0,4);

  > subs(u=2*x,Taylor(exp(u),u=0,4));

  > Taylor(exp(x^2),x=0,4);

  > subs(u=x^2,Taylor(exp(u),u=0,2));

  > subs(u=x^2,Taylor(exp(u),u=0,4));

Note in the second example how substituting into the second order Taylor polynomial for produces the fourth order Taylor polynomial for . In completing the exercises below, you will have to choose the orders carefully to get the two polynomials to match up. In practice, of course, you can just use trial and error.

Here are a few more examples.

Multiplication and Division

If you have the Taylor series for , and you want the Taylor series for something like , you just multiply each term of the series for by . If the leading term for the Taylor series of is for some integer k > 0, you can use division to obtain the Taylor series for for any integer . Some examples follow.

Maple commands for two of the examples above appear below. Note the use of the expand command to actually carry the multiplication or division out. Note also that in the case of a quotient, you might have to increase the order by one or two to get the desired result. This is a quirk of the way Maple generates Taylor series.

  > Taylor(x*sin(x),x=0,6);

  > x*Taylor(sin(x),x=0,5);

  > expand(x*Taylor(sin(x),x=0,5));

  > Taylor(sin(x)/x,x=0,5);

  > Taylor(sin(x),x=0,5)/x;

  > expand(Taylor(sin(x),x=0,5)/x);

Term-by-term integration and differentiation

One of the nice properties of Taylor series is that they can be integrated and differentiated term-by-term. Here is the formal theorem, but the procedures are pretty straightforward.

Exercises

1. Verify that substitution and multiplication work as described above to generate Taylor series (with base point a=0) for the following functions. That is, compare the Taylor polynomials for various orders obtained directly with those obtained by substitution, multiplication, or division.
   1. .
   2. .
   3. .
   4. .
   5. .

2. Verify that the formulas in the theorem make sense. That is, show that differentiating and integrating the series for termwise produce the formulas shown. This probably better done by hand than using Maple.

3. When you differentiate or integrate a series term-by-term, is the convergence of the resulting series changed? That is, does the series resulting from termwise differentiation or integration converge wherever the original series did? Investigate for the functions and .

4. Can you find the sum of the series

   

   if ?