Harmonic Number Calculator
Calculate the nth harmonic number H(n) = 1 + 1/2 + 1/3 + … + 1/n, a fundamental quantity in number theory and analysis.
Range: 1 – 100,000. Whole numbers only
Compute H(n) = Σ 1/k for k=1 to n
Results
H(n)
2.928968254
Approximation (ln n + γ)
2.8798007579
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Harmonic Number Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.
Review your inputs
Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Harmonic Number Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Harmonic Number Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Harmonic Number Calculator
The harmonic numbers are the partial sums of the harmonic series: H(n) = 1 + 1/2 + 1/3 + … + 1/n. Despite their simple definition, harmonic numbers are remarkably important in mathematics and computer science. They appear in the analysis of sorting algorithms (quicksort's expected comparisons involve H(n)), the coupon collector's problem, random permutations, and the distribution of prime numbers. The harmonic series itself diverges — H(n) grows without bound as n increases — but it does so extremely slowly, growing as ln(n) + γ where γ ≈ 0.5772 is the Euler–Mascheroni constant. This slow divergence means that even H(10^100) is only about 230.8. Harmonic numbers connect to the Riemann zeta function (ζ(1) diverges), the digamma function (ψ(n+1) = H(n) − γ), and Stirling numbers. This calculator computes exact harmonic numbers alongside the logarithmic approximation so you can see how accurate the asymptotic formula is.
The Math Behind It
Formula Reference
Harmonic Number
H(n) = Σ (1/k) for k = 1 to n
Variables: n = upper limit
Asymptotic Approximation
H(n) ≈ ln(n) + γ + 1/(2n)
Variables: γ ≈ 0.5772 (Euler–Mascheroni constant)
Worked Examples
Example 1: Compute H(5)
Find H(5) = 1 + 1/2 + 1/3 + 1/4 + 1/5.
H(5) = 137/60 ≈ 2.2833
Common Mistakes & Tips
- !Assuming the harmonic series converges — it diverges, albeit very slowly.
- !Using the approximation ln(n) + γ for small n where the error is significant.
- !Confusing harmonic numbers H(n) with harmonic means.
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
Does the harmonic series converge?
No. The harmonic series diverges, meaning H(n) → ∞ as n → ∞, but it grows very slowly — approximately as ln(n) + 0.5772.
What is the Euler–Mascheroni constant?
γ ≈ 0.5772156649 is defined as the limit of H(n) − ln(n) as n → ∞. It is not known whether γ is rational or irrational.
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