Telescoping Representations of Classical Iterative Methods and Constants

Every convergent sequence $(x_n)$ with limit $L$ satisfies the elementarytelescoping identity $L = x_1 + \sum_{n\ge1}(x_{n+1}-x_n)$. This note takes thatidentity as an organising lens for several classical sequences --- the$(1+1/n)^n$ approximation to Euler's number $e$, its symmetric companion$e+e^{-1}$, the two-parameter family $(1+a/n)^{bn}\to e^{ab}$, theWallis product, and the arithmetic--geometric mean (AGM) --- and, foreach, recordsthe asymptotics of the successive differences $D_n := x_{n+1}-x_n$ together withcheckable error terms. For $(1+1/n)^n$ we give the expansion of$D_n$ through order $n^{-5}$ with exact rational coefficients; for the symmetricsequence we show the differences carry a genuinely different leading constant,$(e-e^{-1})/2$ rather than $e/2$; and for the parametric family we give theleading and next-order coefficients, correcting a form in which the leadingcoefficient degenerates when $a=b$. For the Wallis and AGM sequences we recordthe two-term tail asymptotic (Wallis) and two sharp elementary one-stepinequalities (AGM). Every numeral in the paper is generated by anaccompanying program in exact or arbitrary-precision arithmetic; the finitaryand algebraic statements are additionally checked by a machine-verified Lean~4development, and the asymptotic coefficients are confirmed to high precision byan independent computation. We are explicit throughout about which claims areproved, which are kernel-checked, and which rest on computation.