Five Approaches to Computing e: Derivations, Error Bounds and History

Euler’s number e admits five classical characterisations: as the limit of (1+1/n)^n, as the sum of the reciprocals of the factorials, as the base whose exponential function is its own derivative, as the number whose natural logarithm is 1, and through its regular continued fraction [2;1,2,1,1,4,1,1,6,…]. This paper gives a self-contained account of the five approaches with complete proofs that they define the same number, and for each approach that yields an algorithm it proves an explicit error bound: e-(1+1/n)^n=e/(2n)+O(n^(-2) ), a remainder below 1/(n⋅n!) for the n-th partial sum of the series, and |e-p_k/q_k |<1/q_k^2 for the continued fraction convergents. It shows that the limit is exactly Euler’s method applied to y'=y, that the trapezoidal rule yields the second-order approximation ((2n+1)/(2n-1))^n, and that Newton’s method turns the integral definition into a quadratically convergent scheme. The methods are compared by the work needed per correct digit, and the historical account is corrected against primary and secondary sources.