Elementary proofs of generalized continued fraction formulae for $e$

TL;DR

This paper presents elementary proofs for two generalized continued fraction formulas for e, including a new formula, and discusses the potential for automated verification using computer algebra systems.

Contribution

It provides elementary proofs for known and new continued fraction formulas for e and explores automated verification methods.

Findings

Elementary proofs for two continued fraction formulas for e.

Discovery of a new continued fraction formula for e.

Discussion on automating verification of such formulas using CAS.

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).