Advanced Calculus fi..

434 Advanced Calculus, Fifth Editionare power series with convergence radii r,* and r,*, respectively, 0 < r,* 5 rf, thenwhere -kn = coCn + clCn-1 + ~2Cn-2 + '' ' + cn-1C1 + cnCo.This is simply an application of the Cauchy product rule (Section 6.10) to theabsolutely convergent series for f (x) and F(x).THEOREM 43 Convergent power series can be divided, provided that there isno division by zero; that is, if f (x) and F(x) are given as in Theorem 42 andF(a) = Co # 0, then-for some positive number r;, where the pn satisfy the equations . . 3Cn = poCn + plcn-1 + .-. + pn-lC1 + pnCo. (6.55)The rule (6.55) expresses the fact that the seriespn(x - a)" multiplied bythe series C Cn(x - a)" gives the series C cn(x - a)". A proof of this theoremand a more precise determination of r; require complex variables; this is taken upin Chapter 8. It should be noted that the Eq. (6.55) are implicit equations for thecoefficients pn :co = poco, Cl = poC1 + p,Co, . . .These can be solved in turn to obtain as many coefficients as are desired:it will usually be difficult to obtain a formula for the general coefficient pn.THEOREM 44 A Taylor series with constant term a can be substituted for thevariable x in a Taylor series about x = a; that is, ifhave nonzero convergence radii r,* and r,*, respectively, and Ig(x) - a( < r,* forIx - bl < r ~, where r2 5 rl*, thenwhere the coefficients qn are obtained by collecting terms of same degree.