556 Chapter^8
coefficients an. In fact, either case may occur, as the following examples
show.
Examples 1. If
.f(z)=- tc^1 Log(l-()d(= f~:,
Jo n=llzl < 1the function f has a singularity at z = 1, yet the power series converges for
z = 1. As a matter of fact, it converges absolutely for every z on lzl = 1.
2. If f(z) = Log(l +z) = 2::;'= 1 [(-l)n-^1 /n]zn, .lzl < 1, the function is
singular and the series diverges for z = -1.
At regular points off on the circle lzl = R (if any) the same situation
occurs. For instance, in Example 2 above the function is regular and the
series converges for z = 1, while in the example
- 1/(1 + z) = I::;'= 0 (-lrzn, lzl < 1, the function is regular but the
series diverges for z = 1. It also diverges for every z such that lzl = 1.
The examples above also show that a power series may converge at every
point of its circle of convergence, or it may diverge at every point, or it
may converge at some points and diverge at other points.
However, as mentioned above, by imposing some conditions on the coef-
ficients of the series, a number of theorems have been obtained concerning
the behavior of either the power series or the sum function f(z) on the
circle of convergence. In fact, an extensive literature exists on this topic;
see, for instance, [37], [8], or [17].
In what follows we discuss a few of the main results.
Theorem 8.14 (Picard's Theorem). Consider the power series I: anzn
and suppose that:
- The coefficients an are real nonnegative numbers.
2. an;::: an+l (n = 0,1,2 ... ).
- an -+ 0 as n -+ oo.
Then the series converges at all points of the circle lzl
possibly at z = 1, so its radius of convergence is at least 1.
Proof Let
n+p
Rn,p = L akzk
k=n+l
where p ;::: 1 is an integer. We have
n+p-1
1, except(z - l)Rn,p(z) = -anHZn+l + L (ak - ak+i)zk+^1 + an+pzn+pH
k=n+l