Let $f(x)$ be the expected number of random i.i.d. $[0,1]$ uniform draws needed so that their total exceeds $x$. It's a fairly well-known puzzle that $f(1) = e$ and $f(x) = e^x$ for $x \in (0,1]$.
I'm fairly confident the function can be extended for positive all values of $x$ with the following formula (provable by recurrence, using the fact that $f(x) = \int_{x-1}^x f(u) du$):
$$f(x) = \sum_{i=0}^n \frac{e^{x-i}}{i!}(i-x)^i,\quad \text{where }n = \lfloor x\rfloor$$
But how does one go about studying this function?? To make things a bit easier, I started by considering the sequence
$$f(n) = \sum_{i=0}^n \frac{e^{n-i}}{i!}(i-n)^i$$
Intuitively (from the initial problem), we should have $f(n) \sim 2n$. This appears to be true when computing values numerically. Maybe more surprisingly, numerical values seem to show that $f(n) - 2n \sim 2/3$ ! This I have completely no intuition why it might be the case.
Would greatly appreciate pointers as to how to prove the asymptotic behavior, and maybe some intuitive reason behind the $2/3$ value.
Thanks!