An unexpected limit of expected values

Let t 0. Select numbers randomly from the interval [0,1] until the sum is greater than t. Let α(t) be the expected number of selections. We prove that α(t)=et for 0 t 1. Moreover, . This limit is a special case of our asymptotic results for solutions of the delay differential equation f′(t)=f(t)-f(t-1) for t>1. We also consider four other solutions of this equation that are related to the above selection process.