Theorem of areolar complex continued fraction absolute convergence (Q1346469)

\(b(n)\) \((n \geq 0)\) being a sequence of nonnegative real numbers, the continued fraction \(b(0) + 1/b(1) + 1/b(2) + \dots\) converges if and only if the series \(\sum b(n)\) diverges. The main result of the paper is a simple corollary to this theorem. It is that, \(w(n)\) \((n \geq 0)\) being a sequence of functions analytic over a prescribed domain \({\mathfrak G}, (A)\) the continued fraction \(| w (0 | z) | + 1/ | \overline z w(1 | z) | + 1/ | \overline z w(2 | z) | + \dots\) diverges over \({\mathfrak G}\) if \({\mathfrak G}\) is bounded and the series \(\sum | w(n | z)|\) converges for all \(z\) in \({\mathfrak G}\) while \((B)\) the continued fraction converges over \({\mathfrak G}\) if \({\mathfrak G}\) does not contain the origin and the series diverges for all \(z\) in \({\mathfrak G}\).