@Carl Love Good catch! It's true that the formula is not restricted to n≤365. For n>365, the product automatically contains the factor (365−365)/365=0, so it correctly gives a probability of 0 for no shared birthdays (and therefore a probability of 1 for at least one shared birthday). Thanks for pointing that out.

@sand15 you’re absolutely right that quantities of different dimension (like length, area, volume) are not directly comparable, and measure theory provides many examples where intuition can fail. However, to me, what makes Gabriel’s Horn especially interesting is that both the finite volume and infinite surface area arise from the same curve y=1/x through very similar integrals. The difference comes from how quickly the integrands decay, so I figure the surprise is not just about dimension, but about how small differences in asymptotic behavior determine whether an infinite process produces a finite or infinite result.