Elementary proofs of generalized continued fraction formulae for $e$

Zhentao Lu

arxiv: 1907.05563 · v1 · pith:OT6CIPMNnew · submitted 2019-07-12 · 🧮 math.NT · math.HO

Elementary proofs of generalized continued fraction formulae for e

Zhentao Lu This is my paper

Pith reviewed 2026-05-24 22:50 UTC · model grok-4.3

classification 🧮 math.NT math.HO MSC 11A55

keywords continued fractionselementary proofsegeneralized continued fractionscomputer algebra systems

0 comments

The pith

Two generalized continued fraction formulae for e admit elementary proofs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that e equals the continued fraction 2 + 1/(1 + 1/(2 + 2/(3 + 3/(4 + ⋯)))) and also equals 3 + (-1)/(4 + (-2)/(5 + (-3)/(6 + ⋯))) through direct algebraic manipulations of the exponential series. A sympathetic reader would care because the proofs use only standard series expansions and recursive relations, avoiding convergence tests or special functions. The first identity receives an elementary treatment while the second is presented as newly found, and the note considers whether computer algebra systems can automate checks of similar identities.

Core claim

We prove using elementary methods that e equals the continued fraction 2 + 1/(1 + 1/(2 + 2/(3 + 3/(4 + ⋯)))) and also equals 3 + (-1)/(4 + (-2)/(5 + (-3)/(6 + ⋯))), with the second formula being new.

What carries the argument

The two generalized continued fraction expressions, obtained by matching partial sums of the series for e to recursive fraction relations.

If this is right

Both expressions supply valid representations of e that can be truncated for approximation.
The signed continued fraction provides an alternative form with negative partial numerators.
Elementary techniques suffice to establish these identities without invoking advanced analysis.
Computer algebra systems offer a route to automatic verification of similar continued fraction claims.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same series-matching approach might recover other known continued fraction expansions for e.
The new signed formula could guide searches for continued fraction representations of related constants.
Separate numerical verification of convergence would be a natural next step beyond the algebraic proofs.

Load-bearing premise

The infinite continued fractions converge so that the algebraic manipulations yield equality to e.

What would settle it

Numerical evaluation of successive convergents of either continued fraction that fail to approach the known decimal expansion of e.

read the original abstract

In this short note we prove two elegant generalized continued fraction formulae $$e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\ddots}}}}$$ and $$e= 3+\cfrac{-1}{4+\cfrac{-2}{5+\cfrac{-3}{6+\cfrac{-4}{7+\ddots}}}}$$ using elementary methods. The first formula is well-known, and the second one is newly-discovered in arXiv:1907.00205 [cs.LG]. We then explore the possibility of automatic verification of such formulae using computer algebra systems (CAS's).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper supplies elementary proofs for two continued fraction formulas for e, one classical and one from a prior preprint, but does not rigorously establish convergence for the version with negative terms.

read the letter

The main point is that this is a short note offering elementary proofs for two generalized continued fraction representations of e. The first one is described as well-known, and the second comes from an earlier arXiv paper in computer science. The authors also touch on using computer algebra systems to check such identities automatically. What the paper does is take finite truncations, manipulate them algebraically to match partial sums or expressions involving e, and then suggest the infinite case follows. If the algebra holds, that part is straightforward and avoids special functions or advanced theorems. The soft spot is the passage to infinity. For the equality to be valid, the continued fraction must converge, and its value must be e. The first formula has positive terms, which helps, but the second has negative numerators, so it can oscillate or require different justification. The abstract and the description don't indicate a dedicated convergence proof or error estimate, so that step rests on assumption. The citation pattern is fine; it acknowledges the origin of the second formula. No circularity or invented entities here. This paper is for people who enjoy elementary derivations in number theory and might want to see these particular identities worked out by hand. A reader looking for new results or deep insights into continued fractions won't find much. It could be useful as a teaching example or for someone exploring CAS verification. I don't think it needs peer review. The contribution is too incremental, and the missing convergence detail would likely come up in any review anyway. It can stay as an arXiv note.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to furnish elementary proofs of two generalized continued fraction formulae for e, namely e = 2 + 1/(1 + 1/(2 + 2/(3 + ...))) (a known result) and e = 3 + (-1)/(4 + (-2)/(5 + ...)) (newly discovered in a prior arXiv preprint), together with a brief discussion of automatic verification of such identities via computer algebra systems.

Significance. If the central derivations can be completed with explicit convergence arguments, the elementary proofs would supply accessible derivations for these particular continued-fraction expansions of e, which may be of interest to researchers working on explicit representations of transcendental constants. The CAS-verification remarks constitute a modest computational contribution.

major comments (2)

[Proofs of the two formulae (body of the note)] The proofs proceed by algebraic manipulation of finite truncations or formal expressions that are equated to partial sums involving e, followed by passage to the infinite case; however, no separate argument is given that the continued fractions converge (to the asserted limit). This is load-bearing for both formulae, and especially so for the second, whose negative numerators preclude direct application of the standard positive-term convergence theorems.
[Proofs of the two formulae (body of the note)] No error bounds, monotonicity statements, or appeal to a general convergence criterion for generalized continued fractions are supplied to justify the limiting step; without these, the equality statements remain formal rather than rigorously established.

minor comments (1)

[Abstract] The abstract and the displayed formulae would benefit from explicit notation distinguishing the finite convergents from the infinite continued fraction whose value is claimed to be e.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit convergence arguments. We agree that these are necessary to make the proofs rigorous and will revise the manuscript to supply them.

read point-by-point responses

Referee: [Proofs of the two formulae (body of the note)] The proofs proceed by algebraic manipulation of finite truncations or formal expressions that are equated to partial sums involving e, followed by passage to the infinite case; however, no separate argument is given that the continued fractions converge (to the asserted limit). This is load-bearing for both formulae, and especially so for the second, whose negative numerators preclude direct application of the standard positive-term convergence theorems.

Authors: We agree that a separate convergence argument is required. In the revision we will add explicit proofs: for the first continued fraction we establish that the convergents form a monotonic bounded sequence whose limit must be e; for the second we supply remainder estimates that show the convergents approach e despite the sign changes. revision: yes
Referee: [Proofs of the two formulae (body of the note)] No error bounds, monotonicity statements, or appeal to a general convergence criterion for generalized continued fractions are supplied to justify the limiting step; without these, the equality statements remain formal rather than rigorously established.

Authors: The observation is correct. The revised version will include the missing error bounds and monotonicity statements (or an appeal to a suitable general criterion) so that the limiting step is fully justified for both formulae. revision: yes

Circularity Check

0 steps flagged

No circularity; direct elementary proof against external constant

full rationale

The paper claims to derive the two continued-fraction identities for e by elementary algebraic manipulations on finite truncations that are then passed to the limit. No parameters are fitted to data, no self-citation supplies a uniqueness theorem or ansatz that the present derivations rely upon, and the cited prior arXiv (1907.00205) is invoked only for discovery attribution of the second formula, not as a load-bearing justification for the algebraic steps themselves. The derivation chain therefore remains self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on unstated convergence of the continued fractions and validity of elementary algebraic manipulations.

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