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POWER SERIES

A power series about a, or just power series, is any series that can be written in the form,

where and are numbers. A power series may converge for some values ofxand not for other values ofx.Example: For what values of x does the series converge?THEOREM.For a given power series , there are only 3 possibilities.

The series converges only for = . The series converges for all There is a positive number R such that the series converges if

< and diverges if

>

The number R in case 3 is called the radius of convergence of the power series.

case 1: =

case 2: =

The interval of convergenceof a power series is the interval that consists of all values of x for which

the series converges.

case 1: {a} or = case 2: , case 3: there are four possibilities ,

, + , , + , , + , , + Examples:Find the radius and interval of convergence of the following power series.

1 1 + 3

4

2 5

3 12!

4 6

5 1

4

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Power Series Representations of Functions

Recall that the geometric series is

= provided that |r|

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=

=

= +

+ 1

Theorem: If

=

a a ad f ceece f

> then,

=

= +

+ 1

and both of these also have a radius of convergence of Example:Find a power series representation for the following function and determine its interval of

convergence.

1 = 1

1 2 = 5 3 =

TAYLOR AND MACLAURIN SERIES.

Definition.If f has a power series representation at that is, if

=

< then its coefficient are given by formula

=

! Definition.The power series of the form

=

!

is called the Definition.The particular case of the Taylor series for = is called the

.

=

!

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Definition:The polynomial

=

!

is called the .Examples: Find the Maclaurin series of the following functions.

1 = 2 = 3 = 4 = c 5 = c36 =

7 = 8 = a