Error terms for continued fractions of $e^{1/s}$ and $\sqrt{\frac{v}{u}}\tanh\!\Bigl(\frac{1}{\sqrt{uv}}\Bigr)$

TL;DR

This paper investigates error term decompositions in continued fractions for specific functions like e^{1/s} and a hyperbolic tangent expression, revealing new identities and error analysis techniques.

Contribution

It introduces explicit error term decompositions for continued fractions of e^{1/s} and a hyperbolic tangent expression, expanding understanding of their approximation properties.

Findings

Derived new error term identities for e^{1/s} continued fractions

Established decompositions for a hyperbolic tangent related continued fraction

Connected error sums to the original functions and their properties

Abstract

Many classical identities arise from nothing more mysterious than looking at the same object in two different ways. A number, a function, or a combinatorial object may admit several natural decompositions, and by disassembling it in one way and reassembling it in another, we often obtain unexpected corollaries. Telescoping sums provide a particularly vivid incarnation of this principle: by arranging terms so that successive contributions cancel, one performs a conceptual ``cut-and-paste'' that often admits a clean geometric interpretation. Generating functions offer a complementary perspective. Encoding a problem into a formal power series and then evaluating that series at a prescribed point naturally expresses the same quantity as an infinite (or finite) expansion, and equating these representations yields a wealth of identities. For example, for a real number \(\alpha\) given by its continued fraction expansion $\alpha = [a_0, a_1,a_2,\dots]$, with convergents \(p_n/q_n\) and error terms $E_n := p_n - \alpha q_n$, one can obtain ``additive'' decompositions of the form $\sum_{n\ge-1} a_{n+1}\,\lvert E_n\rvert \;=\; \alpha + 1$, $\sum_{n\ge-1} a_{n+1}\,E_n^{2} \;=\; \alpha$. Thus $\alpha$ and $\alpha+1$ themselves appear as weighted sums of the local approximation errors of their convergents. In this note we explore what such decompositions yield in two explicit cases: the continued fraction \[ e^{1/s} = [1;\,{\overline{(2k-1)s-1,1,1}}]_{k=1}^{\infty} \] and the continued fraction \[ \frac{s}{u}\tanh\!\Bigl(\frac{1}{s}\Bigr) = [\,0;\,\overline{(4k-3)u,\,(4k-1)\tfrac{s^{2}}{u}}\,]_{k=1}^{\infty} \]