204
By elementary algebra we have
1 - zn+l
Sn(z)=l+z+···+zn= ---
1-zprovided that z -:/:- 1.
Chapter41-z 1-z
Obviously, the series converges for z = 0, and for any z such that 0 <
lzl = r'· < 1, we have zn+l -- 0 as n -- oo. In fact, given E > 0, the
inequality lzn+l - OI = rn+l < E is satisfied by taking n > (ln c/ ln r) - 1.
Hence the geometric series converges pointwise in the unit disk lzl < 1,
and its sum is
1
f(z) = lim Sn(z) =
n=oo 1-Z( 4.11-3)Moreover, it converges absolutely in the same disk, since the series of
the absolute values of the terms is the real geometric series
00 00n=O n=O
which converges for 0 ::; r < 1. For any z such that lzl 2: 1 the se-
ries ( 4.11-2) diverges, since this implies that lzln 2: 1, so the general term
does not tend to zero (a sufficient condition for divergence).
Finally, the series converges uniformly on any disk lzl ::; r 1 of fixed
radius r 1 < 1. In fact, on such a disk lzln::; rf, and since I>f converges,
it follows that I: zn converges uniformly on lzl ::; r 1 by the Weierstrass
M-test.
The situation in the general case is somewhat similar to the case of the
geometric series and is summarized in the following theorem.
Theorem 4.19 To every power series l::::"=o anzn there corresponds a real
number R (0 ::; R ::; oo) given by the formula
Rl = limsup \lfanl
n->OO(Cauchy-Hadamard formula) ( 4.11-4)which determines the behavior of the series as follows:- If R = 0, the series converges for z = 0 only.
2. If R = oo, the series converges absolutely for every z E C, and
uniformly on any disk lzl ::; r, 0 < r < oo.
3. If 0 < R < oo, the series converges absolutely for every z such that
lzl < R, and diverges for lzl > R. At a point on the circle lzl = R
the series may converge or diverge. If 0 < r < R, the series converges
uniformly on I z I ::; r.