## 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. Use substitution followed by integration to generate the first three terms in the Taylor series with base point for . Start with the series for .

3. Can you find the sum of the following series?

(Hint - differentiate the series for , then multiply by a power of .)

4. In the background section we only considered multiplication of series by polynomials. Suppose you wanted to generate the Taylor polynomial of order ten with base point for the function . Can you do this by multiplying Taylor polynomials for and ?

William W. Farr
2006-04-10