Harmonic Series Sum Calculator

About this calculator

The harmonic series is the sum 1 + 1/2 + 1/3 + 1/4 + ... + 1/n. Each term shrinks, and it’s tempting to assume the total settles down to a fixed number the way 1 + 1/2 + 1/4 + 1/8 + ... does — but the harmonic series is a classic counterexample: it diverges, growing without bound, just agonizingly slowly. Its partial sum after n terms is closely approximated by ln(n) + γ, where γ ≈ 0.5772 is the Euler-Mascheroni constant, meaning it takes roughly e^100 terms just to pass 100.

This calculator computes the exact partial sum through your chosen n along with the value of the final term (1/n), which is a staple exercise in calculus courses covering series convergence and divergence tests. The harmonic series’ name actually comes from music: the frequencies of overtones in the harmonic series of a vibrating string are proportional to 1, 1/2, 1/3... of the fundamental wavelength, which is why the same 1/n pattern shows up in acoustics and instrument design as well as in analyzing the average-case running time of algorithms like quicksort.