# PP 12.8_1 - Power Series Definition where x is a variable...

This preview shows pages 1–9. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Power Series Definition where x is a variable and the c n ’s are constants called the coefficients of the series. 2 3 1 2 3 ... n n n c x c c x c x c x ∞ = = + + + + ∑ A power series is a series of the form whose domain is the set of all x for which the series converges. 2 1 2 ( ) ... ... n n f x c c x c x c x = + + + + + The sum of the series is a function Example . series power the of domain the Determine n ∑ ∞ = n x ∑ ∑ ∞ =- ∞ = = 1 1 m m n n x x So, we have a geometric series with r = x . By the Geometric Series Test, the series converges for | r = x| < 1 and diverges otherwise. Since all tests available start the sum at n = 1, we need to modify the given series by making m = n + 1. Hence the domain of the power series is the interval (-1, 1). Notation: Since the domain of a power series is usually an interval of real numbers, it is called the interval of convergence of the power series. Thus, the interval of convergence of the series under consideration is (-1, 1). 2 1 2 ( ) ( ) ( ) ... n n n c x a c c x a c x a ∞ =- = +- +- + ∑ Generalization A series of the form is called a power series centered at a or a power series about a Examples 0. at centered is ∑ ∞ = n n x 1. at centered is ) 1 ( 2 ∑ ∞ =- n n n x . at centered is )! 2 ( ) ( π π- + ∑ ∞ = n n n x Comments The sum of a power series resembles a polynomial....
View Full Document

{[ snackBarMessage ]}

### Page1 / 24

PP 12.8_1 - Power Series Definition where x is a variable...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online