Harmonic Number Calculator
Calculate harmonic numbers and explore the harmonic series with step-by-step solutions
Calculate Harmonic Numbers
Enter any positive number (integers give exact results, decimals use approximation)
Harmonic Number Result
Definition: H(n) = 1/1 + 1/2 + 1/3 + ... + 1/n = Σ(k=1 to n) 1/k
Calculation: Direct summation of 5 terms
Euler-Mascheroni constant: γ ≈ 0.5772156649
Step-by-Step Calculation
Sum Formula:
Partial Sums Growth
Notice how the harmonic series grows slowly but without bound
Harmonic Properties
H₁ = 1 is the only integer harmonic number
Harmonic series diverges (grows infinitely)
H(n) ≈ ln(n) + γ for large n
Growth rate is logarithmic
Related to natural logarithm and Riemann zeta function
Famous Harmonic Numbers
Key Constants
Understanding Harmonic Numbers
What are Harmonic Numbers?
The nth harmonic number H(n) is the sum of the reciprocals of the first n natural numbers. It represents the partial sum of the harmonic series up to the nth term.
Mathematical Definition
H(n) = 1/1 + 1/2 + 1/3 + ... + 1/n
= Σ(k=1 to n) 1/k
Key Properties
- •H(1) = 1 is the only integer harmonic number
- •The sequence grows without bound but very slowly
- •Growth rate is approximately logarithmic
Asymptotic Approximation
For large values of n, harmonic numbers can be approximated using the natural logarithm and the Euler-Mascheroni constant.
H(n) ≈ ln(n) + γ + 1/(2n)
where γ ≈ 0.5772... (Euler-Mascheroni constant)
Applications
- 📊Analysis of algorithms (average case complexity)
- 🎵Music theory and harmonic series
- 🧮Number theory and analytic number theory
- 📈Probability theory and statistical analysis
- 🔬Physics (quantum mechanics, statistical mechanics)
Historical Note
Harmonic numbers were studied by ancient mathematicians and are closely related to the divergence of the harmonic series, first proven by Nicole Oresme in the 14th century.