Another way to phrase case (ii) of Theorem 4.15 is to say that the power series converges if and diverges if  .  We call the number the radius of convergence of the power series (see Figure 4.3).  For case (i) of Theorem 4.15, we say that the radius of convergence is zero and that the radius of convergence is infinity for case (iii).

            Figure 4.3  The radius of convergence of a power series.  
                What happens on the boundary circle may be unknown.

Theorem 4.16 (Radius of Convergence).  For the power series function ,  we can find   , its radius of convergence, by any of the following methods:

    (i)    Cauchy's Root Test:        (provided that the limit exists.)  

    (ii)   Cauchy-Hadamard Formula:        (the limit superior always exists.)  

    (iii)  d'Alembert's Ratio Test:        (provided that the limit exists.)

We set    if the limit equals  0,  and    if the  limit equals  .  

Proof.

Proof of Theorem 4.16 is in the book.

Complex Analysis for Mathematics and Engineering