Harmonic Series Calculator

Work out harmonic series instantly with clear inputs, formula shown and shareable results.

About Harmonic Series Calculator

The harmonic series diverges, but slowly: Hₙ ≈ ln n + γ where γ ≈ 0.5772157 is the Euler-Mascheroni constant. The related p-series converges only when p > 1.

Formulas

Harmonic number

Hₙ = Σₖ₌₁ⁿ 1/k ≈ ln n + γ + 1/(2n)

p-series

Σ 1/kᵖ converges if p > 1, diverges if p ≤ 1

Worked example

With the default values below, Harmonic Series Calculator returns Hₙ = 1 + ½ + ⅓ + … + 1/n of 5.18737752. Change any input above and every figure here is recalculated in your browser.

Inputs used

Number of terms naccepts 1–10,000,000: 100
Use this power p for the p-seriesminimum 0: 1

Results

Hₙ = 1 + ½ + ⅓ + … + 1/n: 5.18737752
Estimate ln n + γ + 1/(2n): 5.18738585
Sum of 1/kᵖ for your p: 5.18737752
Final term 1/n: 0.0100000000

Step by step

1Values used
2Harmonic number
3p-series
4Hₙ = 1 + ½ + ⅓ + … + 1/n= 5.18737752
5Estimate ln n + γ + 1/(2n)= 5.18738585
6Sum of 1/kᵖ for your p= 5.18737752
7Final term 1/n= 0.0100000000

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is H₁₀₀?

About 5.1873775, and the asymptotic estimate gives 5.1873775 too — accurate to seven decimals.

Does the harmonic series really diverge?

Yes. Grouping terms into blocks each summing to at least ½ shows the total exceeds any bound, though it takes over 10⁴³ terms to pass 100.