Intro to Analysis in-class_Part_56

Intro to Analysis in-class_Part_56 - 7.4 Power series 117...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 7.4 Power series 117 7.4 Power series 7.4.1 Radius of convergence Definition 7.4.1. A power series is a series of the form a n x n , where x is a variable. The n th term of a power series is f n ( x ) = a n x n (rather than just a n ). a n x n is a family of series, one for each value of x . We are interested in the subfamily corresponding to A = { x R . . . X | a n x n | converges } . Then we can define a function f : A R , f ( x ) = X a n x n . Definition 7.4.2. For any power series a n x n , ! R 0 such that X a n x n converges absolutely for | x | < R, X a n x n diverges for | x | > R. R is the radius of convergence of the power series. (Note: may have R = 0.) By convention, R = iff the series converges x R . Now must validate the assertion of the definition : that such an R exists. This will be shown by computing R explicitly. Theorem 7.4.3. The radius R of a power series c n z n is given by 1 R = limsup n n p | c n | ....
View Full Document

Page1 / 2

Intro to Analysis in-class_Part_56 - 7.4 Power series 117...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online