ABSTRACT
In (4.06) if we replace s by s-I and combine the two equations, then we find that the majorizing coefficients ba also satisfy the simpler recurrence relation
Dividing this by s2bs and letting s -+ a, we find that
which means that the radius of convergence of the series C b , i is p. Therefore, by the comparison test, the radius of convergence of the series (4.04) is at least p. Since p can be arbitrarily close to r, this radius of convergence is at least r. Well-known properties of power series now confirm that the processes of substitution and termwise differentiation used in $4.2 are justified, and hence that the series (4.04) is a solution of (4.01) within Izl< r. This completes the proof.
