But for any two finite numbers, A and B, the chance that C will fall between them if selected from an infinite range is 1/infinity (or 0).
I think you're assuming we pick C uniformly at random from an infinite range, but this is not possible. In general, the claim is not true: consider picking C from a sample of a standard normal, if A = -B = 2 then with >95% C will lie between A and B.
It's equivalent to this much simpler game: "I am going to give you a positive real-number amount of dollars." No matter what I do, you will win money playing this game, guaranteed! Now, how much would you pay to play this game? $0, because for any amount you'd pay to play, I could arbitrarily choose to give you less in winnings.
Like most problems involving infinity, it's unintuitive/paradoxical because it pretends to model a physically plausible scenario but actually doesn't.
The probability that a random number uniformly drawn from the real line is between 42 and 43 is zero in the same way that “the probability that a random real number selected uniformly from the interval [0 1] is pi/4” is zero.
Integral of 1/2^n ... got that no problem.
I took AB Calculus in high school, three semesters of calculus in college, and a bunch of calculus heavy linear algebra. I’m certain that I knew how to integrate that thing at some point... it’s pretty sad that now I can only stare at it blankly and at best lean heavily on Wikipedia to find an answer.
https://en.wikipedia.org/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%...