Harmonic Number Calculator

About this calculator

The nth harmonic number is the sum of reciprocals Hₙ = 1 + ½ + ⅓ + … + 1/n. It grows without bound but very slowly, approximately Hₙ ≈ ln(n) + γ, where γ ≈ 0.5772 is the Euler–Mascheroni constant — a fact famously counter-intuitive to students seeing the harmonic series diverge for the first time despite its terms shrinking to zero.

Computer scientists rely on harmonic numbers to analyze algorithm complexity — they appear in the average-case running time of quicksort, the expected number of trials in the coupon collector's problem, and the analysis of hashing and randomized algorithms. Mathematics students and educators also use them to illustrate series convergence versus divergence. Enter n to get Hₙ to several decimal places.