Where e Hides - Secretary Problem

In optimal stopping problems, marriage problem, Googol game, or the secretary problem, the best strategy is to interview 36.8% of the candidates, and then to take the next candidate that is better than any of the previous ones. 36.8 % is 1/e But why does e show up in the problem at all ? I was a little bit hand wavey about how we get the integral for our percentage. This wikipedia page has a pretty good description of how to get the Sum for a discrete number of options https://en.wikipedia.org/wiki/Secreta... But that page is also hand wavey about how to turn the sum into an integral. This stack exchange page addressed that https://math.stackexchange.com/questi...