POWER SERIES Polynomial of degree n Power Series

POWER SERIES Polynomial of degree n Power Series Example (Polynomial with infinit-degree) Example Power Series Polynomial with infinit-degree

POWER SERIES Polynomial of degree n Power Series Example (Polynomial with infinite-degree) Example Power Series at each x infinite series

POWER SERIES Polynomial of degree n Power Series Example (Polynomial with infinit-degree) Example domain is the set of all x for which the series converges.

POWER SERIES Polynomial of degree n Polynomial centered at a or a polynomial about a Example Polynomial centered at 1 Polynomial centered at 0 Power Series centered at a Example Remark: 1) Maclaurin Series centered at 0

POWER SERIES Power Series centered at a Theorem: there are only three possibilities: 1 The series converges only when 1 2 The series converges for all 2 3 There is a positive number R such that the series 3 converges if There is a positive number R such that converges if diverges if Example Definition: The number R in case 3 is called the radius of convergence of the power series.

POWER SERIES Definition: The number R in case 3 is called the radius of convergence of the power series. Theorem: there are only three possibilities: Radius of convergence Theorem: there are only three possibilities: 1 The series converges only when 1 2 The series converges for all 2 3 There is a positive number R such that the series 3 converges if diverges if Example

POWER SERIES Theorem: there are only three possibilities: Radius of convergence 1 2 3 Example Find the radius of convergence How to find R = radius of convergence 1 Find: 2 Find: 3 Use ratio test: (L is a function of x only) Example Find the radius of convergence

POWER SERIES Final-082

POWER SERIES Final-102

POWER SERIES Theorem: there are only three possibilities: Radius of convergence 1 The series converges only when 2 The series converges for all 3 There is a positive number R such that the series 1 2 3 converges if diverges if Example Remark: case 3 say nothing about the endpoints converges if divg convg divg

POWER SERIES Definition: The number R in case 3 is called the radius of convergence of the power series. Theorem: The interval of convergence is the interval that consists of all values of x for which the series converges. Theorem: there are only three possibilities: Interval of convergence there are only three possibilities: Radius of convergence 1 1 2 2 3 3 Remark: Example endpoint of the interval, that is, , anything can happen—the series might converge at one or both endpoints or it might diverge at both endpoints. Remark: Thus in case 3 there are four possibilities for the interval of convergence:

POWER SERIES How to find R = radius of convergence 1 Find: 2 Find: 3 Use ratio test: (L is a function of x only) How to find interval of convergence 1 Find R: 2 Study convg at endpoints: a+R and a-R

POWER SERIES divg convg Example Find the interval of convergence divg Example (Bessel function of order 0) Find the interval of convergence Example Find the interval of convergence

POWER SERIES Final-101

POWER SERIES Final-092

POWER SERIES Final-081

POWER SERIES Final-081

POWER SERIES Final-102

POWER SERIES Final-092

POWER SERIES Final-092

POWER SERIES Final-082

POWER SERIES DIFFERENTIATION Theorem: 1 has radius of convergence 2 radius of convergence of is R Example: radius of convergence of is 1 radius of convergence of is ? ?

Sec 11. 9 & 11. 10: TAYLOR AND MACLAURIN INTEGRATION Theorem: 1 has radius of convergence Example: 2 radius of convergence of is R radius of convergence of is 1 radius of convergence of is ? ? Remark: Although Theorem says that the radius of convergence remains the same when a power series is differentiated, this does not mean that the interval of convergence remains the same. It may happen that the original series converges at an endpoint, whereas the differentiated series diverges there. (also for integration)

POWER SERIES Operations on Power Series Add: Intersection of interval of convergence Subtrac: Multiplicat: Example: Find the first 3 terms of