This is a math question I have wondered about for around 25 years, and this is my first posting to a math site, so please forgive me if this posting is not up to the typical standards of this site.

Let's look at the set of the probabilities of having at least one occurrence of an event if we make n attempts, where the probability of the event occurring is 1/n. So, for example, if we are looking at the probability of the occurrence of rolling a 4 on a die (1/6), we will actually make 6 attempts. The probability of success in that case (at least one occurrence of a 4 in the 6 attempts) is 1-(5/6^6) = 0.6651 (I have rounded the results.) If we make 15 attempts at something with a probability of its happening (on each separate occasion) being 1/15, then the probability of its occurring at least once in those 15 attempts is 1-(14/15^15) = 0.6447. If we use n=1,000,000, then we get 0.63212. So, we see that we have an asymptotic situation. I have always wondered if there was any other known mathematical significance to this specific approximate number/asymptote. Thanks in advance for reading and for all responses!

Steven