Expected Number of Dice Rolls for the Sum to Reach n

A classic probability result is that the expected number of random summands chosen uniformly from the interval [0, 1] that are needed to sum to at least 1 is e. This result is mainstream enough that it has appeared in general audience books such as [4, p. 137] (a proof can be found in [3]). Here we consider a discrete variation, namely, let X be the number of times an n-sided die has to be rolled so that the sum of the rolls is at least n. Then