Sequences, series, summability Continued fractions and limit processes Theorem

Van Vleck's theorem

Builds on Continued fraction Convergents of a continued fraction Argument of a complex number

Statement

Let $0 < \varepsilon < \pi/2$ and let $a_1, a_2, \dots$ be complex numbers with $\abs{\arg a_n} < \pi/2 - \varepsilon$ for every $n$ . Let $f_n$ be the $n$ -th convergent of the continued fraction $f = \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}}.$ Then:

every $f_n$ is finite and satisfies $\abs{\arg f_n} < \pi/2 - \varepsilon$ ;
the sequences $(f_{2n})$ and $(f_{2n+1})$ of even and odd convergents both converge;
the continued fraction converges, that is, $(f_n)$ has a finite limit, if and only if $\sum_{n \ge 1} \abs{a_n}$ diverges.

Remarks

Item 1 follows by induction from the tail: the open sector $S = \set{z \ne 0 : \abs{\arg z} < \pi/2 - \varepsilon}$ is closed under addition, and $z \mapsto 1/z$ maps $S$ onto itself. Here $\arg$ is the argument with values in $(-\pi, \pi]$ . Example: $a_n = 1$ for all $n$ lies in the sector and $\sum \abs{a_n} = \infty$ , so the continued fraction converges; its value is $\frac{\sqrt5 - 1}{2}$ . If $a_n = 2^{-n}$ , the sum $\sum \abs{a_n}$ is finite, and by item 3 the even and odd convergents tend to different limits. For positive real $a_n$ the theorem contains the classical criterion that $\cfrac{1}{a_1 + \cfrac{1}{a_2 + \cdots}}$ converges exactly when $\sum a_n$ diverges.

Going deeper

For a student

The sector condition keeps all convergents away from $0$ and $\infty$ , so only the oscillation between even and odd convergents can prevent convergence. The divergence of $\sum \abs{a_n}$ is what forces the even and odd limits to meet. Compare the Śleszyński–Pringsheim theorem, where large partial denominators give convergence.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Argument of a complex number concept
Continued fraction definition
Simple continued fraction definition
Convergents of a continued fraction definition
Van Vleck's theorem statement

Tag	`KCCFZTP3MM`
URI	`m2://KCCFZTP3MM`
Kind	statement, theorem
Writing	W0 stub
MSC	40A15, 30B70
Status	This copy of the site has no build data. Run `m2 status KCCFZTP3MM`.
About	Continued fraction
Uses	Continued fraction, Convergents of a continued fraction, Argument of a complex number
Changed	2 November 2026