PowerSeries

PowerSeries - STAT 333 Power Series and Generating...

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: STAT 333 Power Series and Generating Functions Let a ,a 1 ,a 2 ,...,a n ,... be a sequence of real numbers. Define A ( s ) = n =0 a n s n for every real number s for which the above power series converges. Obviously whether or not the power series converges at a particular s depends on the coefficients a n . Every power series does exactly one of the following three things: 1. converges only for s = 0. 2. converges absolutely on a bounded open interval (- R,R ) and fails to converge at any point s with | s | > R . The number R is called the radius of convergence and the interval (- R,R ) is called the interval of convergence. The power series may or may not converge at the boundary points s = R (and if it does converge the convergence may be conditional). 3. converges absolutely for all real s (corresponds to the case R = ). If Cases 2. or 3. hold then we call A ( s ) the generating function of the sequence a ,a 1 ,... . It is necessary to specify the domain of A ( s ) [corresponds to the interval of convergence of the power series] along with its functional form. If Case 1. holds we say the generating function does not exist....
View Full Document

This note was uploaded on 10/24/2010 for the course STAT 333 taught by Professor Chisholm during the Spring '08 term at Waterloo.

Page1 / 3

PowerSeries - STAT 333 Power Series and Generating...

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