Consecutive rolls of a die whose face count increases by $1$ with each roll. The game starts with a dice with (let's say) $1000$ faces. Every roll, the die magically gets another face. So, after $1$ roll it has $1001$ faces, after another $1002$, and so on. Now the question is:

What is the probability of rolling at least one $1$ in the first $x$ rolls?


Compute the complement of the propability to not get a 1, i.e.
$$p(x)=1-\prod_{k=1}^x(1-\frac{1}{999+k})= \frac{x}{999+x}$$
