Large partial quotients are rare with a clean tail -- P(a_n >= k) = log2(1 + 1/k)

For a typical real number the probability that the n-th partial quotient of its continued fraction is at least k has the closed form P(a_n >= k) = log2(1 + 1/k), the tail of the Gauss-Kuzmin law. It telescopes from the per-value law P(a_n = k) = log2(1 + 1/(k(k+2))): the product over j >= k of (j+1)^2 / (j(j+2)) collapses to (k+1)/k, so the tail is log2((k+1)/k) = log2(1 + 1/k). Consequences: P(a_n >= 2) = log2(3/2) ~ 0.585, and the tail decays like 1/(k ln 2) for large k -- heavy enough that E[a_n] diverges (the arithmetic mean of partial quotients is infinite, which is why Khinchin's constant is a geometric mean), light enough that any fixed large value is rare. A worked instance from the continued-fractions chat; a note, not a coordinate.