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Unformatted text preview: ECE 3250 THE zTRANSFORM Fall 2008 1. Power series Well have occasion to consider infinite series of the form X n = c n z n , where z is a complex variable and the c n are given complex numbers. Well want to know whether the series converges for at least some z C and, if so, for exactly what values of z the series converges. The special form of power series makes for a tidy theory of convergence. Fact 1: If R o > 0 is such that the sequence { c n R n o : n } is bounded, then the series X n =0 c n z n converges for every z C satisfying  z  > R o . Similarly, if R o > 0 is such that the sequence { c n R n o : n < } is bounded, then the series 1 X n = c n z n converges for every z C satisfying  z  < R o . Proof: Consider the first assertion. Suppose R o > 0 is such that the sequence { c n R n o : n } is bounded from above in magnitude, say by M > 0. If  z  > R o , then X n =0  c n z n  = X n =0  c n R n o  ( R o /  z  ) n M X n =0  ( R o /  z  ) n = M/ (1 R o /  z  ) , where the last equality holds by geometricseries reasoning since R o /  z  < 1. It follows that the infinite sequence { c n z n } is absolutely summable and is therefore summable by Facts 3 and 7 from the handout on sequences and series. The proof of the second assertion is similar. Fact 2: There exists some z C for which the series X n =0 c n z n converges if and only if there exists some R o > 0 for which the sequence { c n R n o : n } is bounded. Similarly, there exists some z C for which the series 1 X n = c n z n converges if and only if there exists some R o > 0 for which the sequence { c n R n o : n < } is bounded. Proof: The if part is just Fact 1. As for the only if, suppose z o C is such that P n =0 c n z n o (respectively, P 1 n = c n z n ) converges. Then the sequence { c n  z o  n : n } (respectively, { c n  z o  n : n < } ) must be bounded, so taking R o =  z o  proves the assertion(s). 1 2 Suppose now that { c n : n Z } are given complex numbers, and suppose there exists R o > 0 such that the sequence { c n R n o } is bounded. Define R a as follows: R a = inf { R o > 0 : { c n R n o : n } is bounded } . Observe that R a = 0 is possible. Similarly, define R b via R b = sup { R o > 0 : { c n R n o : n < } is bounded } . Observe that R b = is possible. Taken together, Facts 1 and 2 reveal that the series X n =0 c n z n converges for every z C satisfying  z  > R a and diverges for every z satisfying  z  < R a . To see why it diverges for such z , note that if the series converged for some z o with  z o  = R o < R a , then { c n R n o : n } would be bounded, and Fact 1 along with the definition of R a would imply that R o R a , a contradiction. Similarly, the series 1 X n = c n z n converges if  z  < R b and diverges if  z  > R b ....
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