The Halley-Euler method.

摘要

The 'Halley-Euler Method' has been discussed under other names (and only in connection to Leonhard Euler) in the literature: Detlef Laugwitz called it 'Euler's hidden lemma', while McKinzie and Tuckey alternately call it Euler's 'First Hidden Lemma' or the 'Summation Comparison Theorem'. In Euler's applications of this implicit method, it embodies the claim that if the respective coefficients of two power series are infinitely close, then the sums of those power series also are infinitely close. This technique pervades much of Euler's early work with series; in his Introductio, the method plays a central role in his derivations of power series expansions for the trigonometric, logarithmic, and exponential functions. In this study, we trace the origins of the technique, and argue for its first appearance in print in 1695 in an article by Edmond Halley on the logarithmic and exponential series. In addition, we examine one of Laugwitz's major claims, that Euler had anticipated the Cauchy Condition for series convergence as early as 1734; upon careful examination of the relevant work, we find that while Euler recognized that one can distinguish convergent series by the behavior of sums of terms in their tails, the condition as presented by Euler is not correct, and not equivalent to the Cauchy Condition. In light of this, several recent reconstructions of Euler's work on infinite series, specifically those of Laugwitz, and McKinzie and Tuckey, are flawed, in that they posit Euler's knowledge of the convergence condition as a hypothesis. Along the way, much of the early work on constructing power series expansions for functions is highlighted.