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