Série Harmônica parcial H_n
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
About this calculator
The partial harmonic series calculator Hₙ computes the sum of reciprocals of the first n positive integers (1 + 1/2 + 1/3 + ... + 1/n). This series is divergent, meaning it grows indefinitely as n increases. The formula used is the cumulative sum of each term 1/k, where k ranges from 1 to the user-specified value.
This tool is useful in mathematical, statistical, or computational problems, such as algorithm approximations or convergence analysis. For large n, the result approximates the natural logarithm of n plus the Euler-Mascheroni constant (γ ≈ 0.5772).
Warning: for very large n, the calculation may take longer or produce numerical inaccuracies due to computational limits. The series does not converge, so Hₙ always increases, albeit slowly. Avoid extremely high values if speed or precision is critical.
Frequently asked questions
What is the harmonic series?
The harmonic series is the sum of reciprocals of positive integers (1 + 1/2 + 1/3 + ... + 1/n). It is divergent, meaning it grows indefinitely as n increases.
How does the calculator compute Hₙ?
It adds terms 1/k sequentially from k=1 to k=n. For large n, mathematical approximations based on the natural logarithm and Euler-Mascheroni constant are used.
What is this calculator used for?
It solves problems in mathematics, physics, or computer science, such as convergence analysis, algorithm complexity, or function approximations.
Why does the result grow so slowly?
The series diverges logarithmically. For example, H₁₀₀₀ ≈ 7 and H₁₀.₀₀₀ ≈ 9.78.
What if the calculation freezes?
For n > 10⁶, direct computation is slow. Use the approximation Hₙ ≈ ln(n) + γ + 1/(2n) for faster results.