1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

7 .2 • TAYLOR SERIES REPRESENTATIONS 263

• EXAMPLE 7.6 Use the Cauchy product of series to show that
  1 00
  --- 2 = L (n + 1) z", for z E Di(O).
  (1-z) n=O
  00
  Solution We let f (z) = g(z) = 1 ~. = I: z", for z E D1 (0). In terms of
  n = O
  Theorem 7.6, we have a.,= bn = 1, for all n, a.nd thus Equation (7-17) gives


• -^1 2 =h(z)=f(z)g(z)= L ""(" L:akbn- k ) z"= L^00 (n+l)z".
  (1 -z) n=O k=O n=O