Harmonic Number Calculator
Sum of Reciprocals
Calculate the nth harmonic number H(n) = 1 + 1/2 + 1/3 + ... + 1/n with detailed steps and approximations.
Calculator
Harmonic Number Formula:
H(n) = 1 + 1/2 + 1/3 + ... + 1/n = Ξ£(1/k) for k=1 to n
About Harmonic Numbers
Harmonic numbers are the sum of reciprocals of the first n natural numbers.
Formula:
H(n) = Ξ£(1/k) for k=1 to n
Approximation:
H(n) β ln(n) + Ξ³
where Ξ³ β 0.5772 (Euler-Mascheroni constant)
Features
- Exact calculation up to 10,000
- Step-by-step breakdown
- Logarithmic approximation
- Error analysis
- Quick example values
- High precision (10 decimals)
- Copy results
Common Values
H(1):1.0000
H(2):1.5000
H(5):2.2833
H(10):2.9290
H(100):5.1874
H(1000):7.4855
Properties
Growth:
H(n) grows logarithmically
Divergence:
H(n) β β as n β β
Recurrence:
H(n) = H(n-1) + 1/n
Connection:
Related to natural logarithm
Applications
- Algorithm analysis (complexity)
- Probability theory
- Number theory
- Combinatorics
- Computer science
- Physics (statistical mechanics)
- Music theory (harmonic series)
- Information theory
Mathematical Facts
Interesting Properties:
Euler-Mascheroni Constant: Ξ³ = lim(H(n) - ln(n)) as nββ
Never Integer: H(n) is never an integer for n > 1
Asymptotic: H(n) β ln(n) + 0.5772 for large n
Harmonic Series: The infinite series Ξ£(1/n) diverges
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