A relationship between the modified Euler method and e

Abstract

Approximating solutions to the differential equation dy/dx=f(x,y) where f(x,y)=y by a generalization of the modified Euler method yields a sequence of approximates that converge to e. Bounds on the rapidity of convergence are determined, with the fastest convergence occuring when the parameter value is 12, so the generalized method reduces to the standard modified Euler method. The situation is similarly examined when f is altered.