---
title: Worpitzky Convergence Disk
url: https://www.emergentmind.com/topics/worpitzky-convergence-disk
type: topic
---

# Worpitzky Convergence Disk

The Worpitzky convergence disk is a rigorous analytic region in the complex plane that prescribes absolute convergence and geometric control of all approximants of a limit-periodic continued fraction. It arises from the Worpitzky criterion, which provides explicit quantitative bounds for the convergence and location of convergents of continued fractions with suitably bounded parameters. In the context of polynomial continued fractions representing values such as $π/4$, as recently formalized in connection with the Ramanujan Machine conjectures, the Worpitzky disk plays a central role in establishing not only convergence but also the symbolic minimality and uniqueness of the coefficient structures underlying these analytic identities [2601.08461].

## 1. The Worpitzky Convergence Criterion

Let $F_∞ = b_0 + K_{n=1}^∞(a_n/b_n)$ denote a continued fraction with nonzero denominators $b_n$. The Worpitzky parameters are defined as $ρ_n := a_n/(b_n b_{n-1})$, with $b_0$ normalized to $1$. The Worpitzky criterion asserts that if for all $n \geq 1$,
$$
|ρ_n| \leq R < 1/4,
$$
then:
- The continued fraction converges absolutely;
- For every $N$, the tail $T_N := K_{n=N}^∞(a_n/b_n)$ and every finite convergent lie in the closed disk centered at the origin of radius
$$
r = \frac{1 - \sqrt{1 - 4R}}{2};
$$
- The Worpitzky disk $D(r)$ is strictly contained in the open unit disk, providing explicit geometric error control.

These properties ensure uniform geometric decay of truncation errors and absolute convergence of the continued fraction whenever the Worpitzky bound is respected [2601.08461].

## 2. Analytic Realization in Ramanujan Machine Conjectures

A recently formalized example concerns the continued fraction:
$$
F_∞ = 0 + K_{n=1}^∞\left(\frac{a_n}{b_n}\right)
$$
with $b_0=0$, $b_n = -(3n-2),\, n\geq 1$; $a_1=a_2=1$, $a_n = -(n-1)(2n-5)$ for $n\geq 3$. Analytically, this structure is equated with the ratio of contiguous Gaussian hypergeometric functions,
$$
-\frac{π}{4} = \mathcal{R}(1/2,0,1/2; -1)
$$
where
$$
\mathcal{R}(a,b,c;z) := \frac{{}_2F_1(a,b+1;c+1;z)}{{}_2F_1(a,b;c;z)}
$$
and for $(a,b,c)=(1/2,0,1/2)$, $z=-1$, the hypergeometric function reduces to the classical alternating series for $π/4$: ${}_2F_1(1/2,1;3/2;-1) = \sum_{n=0}^∞ \frac{(-1)^n}{2n+1}$ [2601.08461].

## 3. Disk Radius and Stability Analysis

For this class, the key parameter is
$$
ρ_n = \frac{a_n}{b_n b_{n-1}} = \frac{-(n-1)(2n-5)}{(3n-2)(3n-5)}, \quad n\geq 3.
$$
Taylor expansion as $n\to\infty$ yields $ρ_n = -2/9 + O(1/n)$, so $|ρ_n| \leq 2/9 < 1/4$ for all $n \geq 3$. The supremum $R=2/9$ then gives, via the Worpitzky formula,
$$
r = \frac{1 - \sqrt{1 - 4R}}{2} = \frac{1 - \sqrt{1 - 8/9}}{2} = \frac{1 - 1/3}{2} = \frac{1}{3}.
$$
Consequently, every tail $K_{n=N}^∞(a_n/b_n)$ and all finite convergents $S_N$ satisfy $|S_N| \leq 1/3$. As $1/3 < 1/2$, absolute convergence is geometric with error decay rate
$$
σ = \frac{1 - \sqrt{1 - 4 \cdot (2/9)}}{1 + \sqrt{1 - 4 \cdot (2/9)}} = \frac{1}{2}.
$$
The absolute bound $|ρ_n| < 1/4$ strictly contains all convergents within the disk $|w| \leq 1/3$, the Worpitzky convergence disk for this system.

## 4. Minimality of Integer Coefficient Realizations

The coefficients $a_n$ and $b_n$ for $n\geq 3$ arise via the unique equivalence transformation $r_n = -(3n-2)$ applied to the base Gaussian continued fraction coefficients $b_n^*=1$, $a_n^* = -(n-1)^2/[4(n-1)^2-1]$. The integer realization is symbolically minimal: no further common integer factor can be removed from all $(a_n, b_n)$ while preserving the limiting ratio $L=-2/9$, and any reduction in polynomial degree for $a_n$ would force $|L| \not< 1/4$, thereby destroying convergence. Specifically, the boundary data $a_1=a_2=1$ are the minimal integer initial values selecting the negative branch $-π/4$ [2601.08461].

## 5. Structural Implications and Absolute Convergence

The containment of all approximants and partial sums within the closed Worpitzky disk $D(1/3)$ provides uniform geometric error control and establishes that the associated limit-periodic continued fraction converges absolutely to $-π/4$. This result demonstrates the nontrivial analytic structure of continued fractions discovered by algorithmic induction, situating them as explicit consequences of classical hypergeometric transformation theory rather than as isolated numerical phenomena.

## 6. Table: Parameters and Constraints in Worpitzky Disk Analysis

| Parameter                | Value/Formulation                               | Condition                                  |
|--------------------------|-------------------------------------------------|--------------------------------------------|
| $ρ_n$                    | $\dfrac{a_n}{b_n b_{n-1}}$                     | $|ρ_n| \leq 2/9 < 1/4$ for all $n\geq 3$   |
| Worpitzky disk radius $r$| $\dfrac{1 - \sqrt{1 - 4R}}{2}$                  | $R = 2/9 \implies r = 1/3$                 |
| Tail and convergent bound| $|S_N| \leq r$                                  | $r = 1/3 < 1/2$                            |
| Error decay rate $σ$     | $\dfrac{1 - \sqrt{1 - 4R}}{1 + \sqrt{1 - 4R}}$  | $σ = 1/2$ for $R=2/9$                      |

All such findings are direct corollaries of the Worpitzky criterion as applied to the specific continued fraction system, confirming that algorithmically discovered analytic identities for constants such as $π/4$ are deeply subsumed under the classical analytic theory of continued fraction convergence [2601.08461].

Source: https://www.emergentmind.com/topics/worpitzky-convergence-disk