Theorem: Pade[1/1](e^x) = (2+x)/(2-x) agrees with Taylor series through O(x^2)

Theorem (Pade-resummation exactness, diagonal case): the [1/1] Pade approximant of exp(x), defined as the unique rational function P_1(x)/Q_1(x) with P_1, Q_1 of degree 1 whose Taylor series matches exp(x) through order x^2, is P/Q =…