Maclaurin and Taylor Series

1. Maclaurin and Taylor Series • Review • We defined: • The nth Maclaurin polynomial for a function f as • The nth Taylor polynomial for f about x = x0 as

2. Maclaurin and Taylor Series

3. Example • Find the Maclaurin series for (a) (b) (c) (d) Solution (a) We take the Maclaurin polynomial and extend it

4. (b)We take the Maclaurin polynomial and extend it. (c)

11. Example • Find the Taylor series for 1/x about x = 1. • In the previous lecture we have found thenth Taylor polynomial for 1/x about x = 1 is • Thus, the Taylor series for 1/x about x = 1 is

12. Power Series in x • If are constants and x is a variable, then a series of the form is called a power series in x. Some examples are

13. Radius and Interval of Convergence • If a numerical value is substituted for x in a power series then the resulting series of numbers may either converge or diverge. This leads to the problem of determining the set of x-values for which a given power series converges; this is called its convergence set. • Observe that every power series in x converges at x = 0. In some cases, this may be the only number in the convergence set. In other cases the convergence set is some finite or infinite interval containing x = 0.

14. Radius and Interval of Convergence

15. Radius and Interval of Convergence

16. Example • Find the interval of convergence and radius of convergence of the following power series. (a) (b)

28. Power Series in x – x0 If xo is a constant and if x is replaced by x – xo in the power series expansion, then the resulting series has the form This is called a power series in x – xo. Some examples are More generally, the Taylor series is a power series in x – xo.

29. Radius and interval of convergence of a Power Series in x – x0

30. Radius and interval of convergence of a Power Series in x – x0

31. Example • Find the interval of convergence and radius of convergence of the series