Harmonic Number Calculator
Free online harmonic number calculator. Enter your values and get instant results with the formula, step-by-step working, a worked example and FAQs — everything runs locally in your browser on Calcton, no signup required.
中文版：调和数在线计算器
调和数 H_n = 1 + 1/2 + … + 1/n：发散得最慢的著名级数，H_n − ln n → 欧拉常数 γ ≈ 0.5772156649。
发散得最慢的级数
H_n = ln n + γ + 1/(2n) − 1/(12n²) + 1/(120n⁴) − …；H_n = ψ(n+1) + γ
调和级数前 10^43 项之和还不到 100——增长慢到需要天文数字才「看起来」线性。应用遍地：优惠券收集问题期望 = n·Hₙ、快速排序平均比较次数 = 2(Hₙ₊₁ − 1)、堆垛书本最大悬挑 = ½Hₙ（调和级数发散意味着能悬出任意远）、算法复杂度里 Σ1/k 处处以 Hₙ 出场。大 n 用渐近式计算到机器精度只需 5 项；整数 H_n 仅在 n = 1 时为整数（Theisinger 定理）。
How to use the Harmonic Number Calculator
- Fill in the inputs with your own numbers — every field shows what it expects.
- The result updates instantly as you type; no "Calculate" click needed.
- Read the result card for the main figure, the step-by-step formula, and practical notes.
Frequently asked questions
Is the Harmonic Number Calculator free?
Yes — all math calculators on Calcton are free, with no sign-up required. Your inputs are processed in your browser and never sent to a server.
Is there a Chinese version of this tool?
Yes. Use the Chinese version link above — it is the same calculator with the interface in Simplified Chinese.
References
Formulas and benchmarks used by this calculator follow these published sources.
KevinContent LeadCalcton Editorial Team · Last updated 2026-09-26
Kevin maintains Calcton's formula verification and content: every calculator cites the standard it implements, worked examples are recomputed programmatically, and health benchmarks follow WHO and international guidelines.
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