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.
Joshua Bald (Tue,) studied this question.