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February 9, 20260 citationsOpen Access

Higher-Order Cancellation in Exponential Approximants: An Explicit O(n−4) Telescoping Family

JBJoshua Bald

Key Points

  • The aim is to study convergence rates of elementary sequences approaching exponential constants through telescoping identities.
  • Analyzed elementary sequences whose differences decay rapidly.
  • Defined telescoping order based on the decay of successive differences.
  • Utilized parametrized families and asymptotic expansions to connect decay rate to parameters.
  • Identified sequences that exhibit telescoping order k with specific decay rates.
  • Established relationships between the decay of differences and algebraic constraints on parameters.
  • Demonstrated explicit O(n−4) behavior in the families studied.

Abstract

We study elementary sequences (xn) converging to exponential constantsec whose successive differences form rapidly decaying telescopingtails. For any convergent sequence (xn) → L, the telescoping identityL = x1 +P∞ n=1 Dn (where Dn := xn+1 − xn) allows us to study convergencerates via the decay of differences. We say (xn) has telescoping order k if Dn = Θ(n−(k+1)).Working with parametrized families xn = f(1/n)g(n) where f(t) = 1+α1t+α2t2 +· · · and g(n) = βn+γ, we use asymptotic expansions of log xn to relate the decay rate of Dn to algebraic constraints on the parameters.

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Cite This Study

Joshua Bald (2025) studied this question.

synapsesocial.com/papers/69897a35f0ec2af6756e899ehttps://doi.org/10.5281/zenodo.18520417
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