I am trying to calculate the sum of a series as shown below:
I have arrived here as an extension of a problem surrounding the expected number of guesses it would take for an individual to guess the birthday of any person in a group, where there are s different possible birthdays and n people in the group. Guesses cannot repeat, but more than one person may share the same birthday.
I think this series represents the odds of guessing an "occupied" birthday in the first 1,2,3... birthdays, but am having trouble getting an equation in terms of s, n, and x that states the overall chances of having guessed right after x guesses, so that I can set it equal to 0.5 and solve for x to find the expected number of guesses.
However, I am really stuck and cannot figure it out. Any help is greatly appreciated.