Beer!

Lewis is singing the familiar song, " $N$ bottles of beer". He starts off by singing " $N$ bottles of beer on the wall, $N$ bottles of beer, take one down, pass it around, $N-1$ bottles of beer on the wall", and continues to sing this phrase, counting down by one bottle each time, until he reaches $0$ . However, Lewis is very forgetful, and every time he completes a phrase, there is a chance that he starts over from the beginning of the whole song. Specifically, after any phrase that Lewis sings, there is a $\frac{1}{N}$ chance that his next phrase begins with " $N$ bottles of beer on the wall". Note that this only applies to phrases that would require Lewis to sing again. This means that after Lewis says " $0$ bottles of beer on the wall", he will stop singing. Let $\alpha$ denote the expected number of phrases Lewis sings before finishing the song. Find $\lim_{N \rightarrow \infty} \frac{\alpha}{N}$ to the nearest thousandth.

Answer format: round to 3 decimal places.