Harmonic Series Calculator
Calculate harmonic numbers H_n and generalized harmonic series with power p.
Input
H_n^(p) = Σ 1/k^p for k=1 to n
Series Definition
H_n = 1 + 1/2 + 1/3 + ... + 1/n
H_100
5.1873775176
Asymptotic Approximation
5.1873775176
H_n ≈ ln(n) + γ + 1/(2n)
Error: 8.3330e-11
Convergence
Diverges
Euler-Mascheroni γ
0.57721566
Terms to Reach Value
H_n = 5: n ≈ 83
H_n = 10: n ≈ 12,367
H_n = 15: n > 1M
H_n = 20: n > 1M
First Terms
| k | 1/k | Partial Sum |
|---|---|---|
| 1 | 1.000000 | 1.000000 |
| 2 | 0.500000 | 1.500000 |
| 3 | 0.333333 | 1.833333 |
| 4 | 0.250000 | 2.083333 |
| 5 | 0.200000 | 2.283333 |
| 6 | 0.166667 | 2.450000 |
| 7 | 0.142857 | 2.592857 |
| 8 | 0.125000 | 2.717857 |
| 9 | 0.111111 | 2.828968 |
| 10 | 0.100000 | 2.928968 |
Harmonic Series
Properties
- H_n diverges as n → ∞ (for p = 1)
- H_n ~ ln(n) + γ for large n
- For p > 1, converges to Riemann zeta ζ(p)
Special Values
- ζ(2) = π²/6 ≈ 1.6449
- ζ(4) = π⁴/90 ≈ 1.0823
- γ ≈ 0.5772 (Euler-Mascheroni)