Harmonic Series Calculator — Partial Sums
Compute how far the harmonic series 1 + 1/2 + 1/3 + … has grown after n terms, and find out how many terms you need before the running total first exceeds a given value.
1 + 1/2 + 1/3 + … + 1/n (exact)
How does this calculator work?
The harmonic series 1 + 1/2 + 1/3 + … grows as ln(n) + 0.5772. After 10 terms the sum is ≈ 2.93; after 100 it is ≈ 5.19; after 1 000 it is ≈ 7.49. Though every term added increases the sum, the series diverges — just very slowly. To exceed a target S, you need roughly e^(S − 0.5772) terms.
Formula
How this is calculated
The harmonic series is the sum 1 + 1/2 + 1/3 + 1/4 + ⋯, where each term is the reciprocal of its position. This calculator computes the partial sum H(n) — the sum of the first n terms — by direct addition, and plots how the running total grows with each added term.
Despite the terms shrinking toward zero, the harmonic series diverges: if you keep adding terms forever, the sum grows without bound. The growth is famously slow — it grows approximately as ln(n) + γ, where γ ≈ 0.5772 is the Euler–Mascheroni constant. To get the sum above 10, you need about 12,367 terms; to exceed 20, about 272 million terms. In practice the partial sums look nearly frozen once n is in the hundreds.
The "terms to exceed target" feature answers questions like "how many of the first reciprocals must you add to surpass a budget or threshold?" It searches up to 10 million terms (covering sums up to about 16.1) and reports the count. Beyond that, use the approximation: n ≈ e^(target − γ).
Frequently asked questions
Going to zero is necessary but not sufficient for convergence. The harmonic series diverges because the terms decrease too slowly. A classic grouping proof shows that you can always extract another group summing to at least 1/2: 1/3 + 1/4 > 1/2; 1/5+1/6+1/7+1/8 > 1/2; and so on, giving infinitely many 1/2 contributions.
You need to sum approximately 12,367 terms. To exceed 20 requires about 272 million terms. The number of terms needed to exceed S grows as e^(S − γ), where γ ≈ 0.5772, so each extra unit of sum costs roughly 2.72 times more terms.
In the "coupon collector" problem — if a cereal box contains one of n equally likely prizes, the expected number of boxes to complete the set is n × H(n). For n = 10 prizes, that is 10 × 2.929 ≈ 29 boxes. The series also models the expected number of unique web links clicked before revisiting one.
Also known as
TG we-Calculate Editorial Team. (2026). Harmonic Series Calculator — Partial Sums [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/harmonic-series-calculator
TG we-Calculate Editorial Team. "Harmonic Series Calculator — Partial Sums." TG we-Calculate. 2026. https://we-calculate.com/calculator/harmonic-series-calculator.
TG we-Calculate Editorial Team, "Harmonic Series Calculator — Partial Sums," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/harmonic-series-calculator
@misc{wecalculate_harmonic_series_calculator, title = {Harmonic Series Calculator — Partial Sums}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/harmonic-series-calculator}}, year = {2026}, note = {TG we-Calculate} }
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