Sequences which converge to e: New insights from an old formula

One of the most fundamental results in calculus was the discovery of the mathematical constant e = 2.718... by Jacob Bernoulli. Remarkably, new definitions of e are still being discovered, in part due to renewed interest at the advent of modern computing and the quest for more digits. In this work we review recent discoveries of sequences which tend to e, and propose a systematic approach for producing such sequences. In doing so, we establish several classes of sequences, and their generalizations. Our methods use only basic tools of calculus and numerical analysis, such as series expansions and Padé approximants. Numerical results demonstrate that our new sequences rapidly converge to e.