High-Order Telescoping Sequences for Function Approximation: A Framework Using Optimized Exponential Approximants

We develop a framework for constructing telescoping sequences that approximatea class of analytic functions --- those built from exponentials by finite sumsand products --- at a fixed third-order convergence rate. The engine is afamily of optimized exponential approximants Eₙ(±)(c)→ eᶜwhose successive differences decay as O(n⁻⁴) uniformly on compact sets;these are embedded in classical identities (the integral representation oflog(1+x), Euler's formula for the trigonometric functions) and closed underfinite algebraic operations, so that any function built from exponentialsinherits the same O(n⁻⁴) difference decay. We give the exact leadingcoefficient of the exponential engine's difference,Eₙ₊₁(±)(c)-Eₙ(±)(c) = c(3∓2√3)/24· eᶜ\,n⁻⁴+ O(n⁻⁵), derived symbolically and confirmed to ten digits byRichardson-extrapolated high-precision computation, and we formalise the exactalgebraic identities underlying it in Lean~4. A complexity analysis gives theterm count N~ε-1/k (here $k=3$) for accuracy ε;we are candid that this algebraic rate is inferior to Chebyshev/spectralmethods on bounded smooth domains, and identify the contexts where the methodis nonetheless useful: unbounded domains, $O(1)$ memory, no linear solves, andnatural parallelism. Every numeral in the paper is regenerated by anaccompanying program, and we distinguish throughout between kernel-checkedalgebra, high-precision-confirmed asymptotics, and heuristic remarks.