Harmonic Number Calculator
Calculate harmonic numbers H_n
How to Use This Calculator
Enter Term Number
Input a positive integer (1, 2, 3, ...) to find the nth harmonic number.
Click Calculate
Press the "Calculate" button to see the harmonic number result.
Review Result
See the harmonic number H_n displayed with high precision.
Formula
H_n = 1 + 1/2 + 1/3 + ... + 1/n = Σ(k=1 to n) 1/k
Where:
- H_n = nth harmonic number
- n = positive integer
- Σ = summation (sum of terms)
Example: Find H₅
H₅ = 1 + 1/2 + 1/3 + 1/4 + 1/5
H₅ = 1 + 0.5 + 0.3333... + 0.25 + 0.2
H₅ ≈ 2.2833
About Harmonic Number Calculator
The harmonic number H_n is the sum of the reciprocals of the first n positive integers. It's an important concept in mathematics, appearing in various areas including number theory, analysis, and algorithm complexity.
Key Properties
- The nth harmonic number grows approximately as ln(n) + γ (Euler-Mascheroni constant)
- H_n diverges to infinity as n approaches infinity
- Harmonic numbers appear in the analysis of algorithms and data structures
- They're used in probability theory and number theory
First Few Harmonic Numbers
- H₁ = 1
- H₂ = 1.5
- H₃ ≈ 1.8333
- H₄ ≈ 2.0833
- H₅ ≈ 2.2833
- H₁₀ ≈ 2.929
Frequently Asked Questions
What is a harmonic number?
A harmonic number H_n is the sum of the reciprocals of the first n positive integers: H_n = 1 + 1/2 + 1/3 + ... + 1/n.
Do harmonic numbers converge?
No, harmonic numbers diverge to infinity. However, they grow very slowly (logarithmically).
Where are harmonic numbers used?
Harmonic numbers appear in algorithm analysis (average case complexity), probability theory, and mathematical series.
What is the relationship to the natural logarithm?
H_n ≈ ln(n) + γ, where γ (gamma) is the Euler-Mascheroni constant (≈ 0.5772). This approximation improves as n increases.
Can I calculate very large harmonic numbers?
Yes, but be aware that very large values may take longer to compute and the harmonic series grows very slowly (logarithmically).