Exact Harmonic Number Calculator

Add reciprocals from one through n using a common denominator and reduce the numerator and denominator after every step. The finite harmonic number is not the divergent infinite harmonic series. Inputs remain whole numbers inside the displayed domain, and the page never rounds a decimal into an accepted integer. This keeps the reported result tied to one explicit mathematical definition rather than an unstated convention.

H(n) grows slowly as terms are added; the exact fraction is authoritative and the decimal is a display approximation. Add the same reciprocals with a common denominator and reduce by their greatest common divisor. Reuse the displayed recurrence, congruence, or counting identity with the same inputs. The verification evidence checks this calculator's result, but it does not establish assumptions or conclusions outside the stated mathematical domain.

The limit n = 20 keeps intermediate numerator and denominator arithmetic inside exact safe-integer storage. The conservative limit preserves both numerator and denominator exactly. A finite limit prevents browser stalls and avoids presenting an unsafe floating-point integer as exact. Work beyond that limit belongs in arbitrary-precision software with explicit resource controls and independent validation.