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 \\emph{optimized exponential approximants} $E_n^{(\\pm)}(c)\\to e^c$whose successive differences decay as $O(n^{-4})$ uniformly on compact sets;these are embedded in classical identities (the integral representation of$\\log(1+x)$, 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,$E_{n+1}^{(\\pm)}(c)-E_n^{(\\pm)}(c) = c(3\\mp2\\sqrt3)/24\\cdot e^{c}\\,n^{-4}+ O(n^{-5})$, 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 $N\\sim\\varepsilon^{-1/k}$ (here $k=3$) for accuracy $\\varepsilon$;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.