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Maclaurin and Taylor Series. Review We defined: The nth Maclaurin polynomial for a function f as The nth Taylor polynomial for f about x = x 0 as. Maclaurin and Taylor Series. Example. Find the Maclaurin series for (a) (b) (c) (d)
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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
Example • Find the Maclaurin series for (a) (b) (c) (d) Solution (a) We take the Maclaurin polynomial and extend it
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
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
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.
Example • Find the interval of convergence and radius of convergence of the following power series. (a) (b)
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.
Radius and interval of convergence of a Power Series in x – x0
Radius and interval of convergence of a Power Series in x – x0
Example • Find the interval of convergence and radius of convergence of the series