The Stern-Stolz theorem is a divergence criterion for a simple continued fraction whose partial denominators
have a finite sum of absolute values. Let 
be complex
numbers. If
 |
(1)
|
then the continued fraction
 |
(2)
|
diverges in the ordinary sense, even when convergence to 
is allowed on the Riemann
sphere (Bowman and Mc Laughlin 2007, Cor. 1).
Let 
be its 
th
convergent, with 
, 
, 
, and 
. The recurrence relations
are 
and 
.
The numerators and denominators
of the even and odd convergents have finite limits,
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
,
1. Taking the limit in the continued
fraction fundamental recurrence relation gives
 |
(5)
|
Consequently, the even and odd subsequences converge to different points 
and 
on the Riemann sphere.
A zero 
denotes the point at infinity. These two distinct subsequential
limits prevent ordinary convergence.
For example, 
with 
satisfies the hypothesis since 
. Thus, even with positive partial denominators, the continued
fraction diverges. This does not contradict convergence of a regular
continued fraction, which requires its partial
denominators after 
to be positive integers.
