Stern-Stolz Theorem

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.