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Formula used
The standard harmonic number adds reciprocals from one through n. Its growth is unbounded, but extremely slow.
The generalized form raises every denominator to an order m. Orders above one approach corresponding Riemann zeta values.
Euler–Maclaurin corrections provide fast estimates for very large indices. Higher selected orders usually reduce approximation error.
How to use this calculator
- Select standard, generalized, alternating, range, comparison, or inverse mode.
- Enter the index, order, range, comparison list, or target value.
- Choose precision, notation, rounding, approximation, table, and chart settings.
- Press the calculation button to generate the result above the form.
- Copy, print, share, or export the completed calculation.
Mathematical notes and applications
Slow logarithmic growth
Harmonic numbers increase without reaching a finite limit. However, doubling n adds only about ln(2). This makes their growth much slower than linear sequences.
Euler–Mascheroni constant
The difference between Hₙ and ln(n) approaches γ. This constant appears throughout analysis, number theory, probability, and asymptotic estimates.
Algorithms and probability
Harmonic numbers describe average behavior in many randomized algorithms. They also appear in coupon collection, records, sorting analysis, and occupancy models.
Exact arithmetic limits
Exact harmonic fractions develop very large numerators and denominators. The calculator uses GMP when available and applies conservative fallback limits otherwise.
Frequently asked questions
What is the first harmonic number?
H₁ equals 1 because the sum contains only 1/1.
Do harmonic numbers converge?
The standard sequence diverges. Generalized harmonic numbers converge as n approaches infinity when m exceeds one.
Why can exact fractions become expensive?
The reduced denominator grows rapidly. Arbitrary-precision integer operations therefore require increasing memory and processing time.
What does compensated summation improve?
It reduces floating-point error accumulated while repeatedly adding small terms to a larger partial sum.
Can this calculator solve for n?
Yes. Reverse mode finds the smallest index whose harmonic value reaches or exceeds a selected target.