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I was told this problem a while ago, and recently someone explained the answer to me, which I didn't understand; could someone please explain in layman's terms (ish)?

You have a die with $n$ sides. Each side is numbered - uniquely - from $1$ to $n$, and has an equal probability of landing on top as the other sides (i.e. a fair die). For large $n$ (I was given it with $n = 1,000,000$), on average how many rolls does it take to achieve a cumulative score of $n$ (or greater)? That is, when you roll it, you add the result to your total score, then keep rolling and adding, and you stop when your score exceeds or is equal to $n$.

The cool thing about this problem: apparently, the answer is $e$. I would like to know exactly how this is derived.