Soma Harmônica Hₙ
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
Hₙ
About this calculator
The Harmonic Sum Calculator Hₙ computes the sum of the reciprocals of the first n natural numbers. This series, defined as Hₙ = 1 + 1/2 + 1/3 + ... + 1/n, is known as the harmonic series. Despite growing slowly, it diverges to infinity as n increases. This tool allows quick calculation of the exact or approximate value for any positive integer n.
The calculation is based on the definition of the harmonic series: iteratively summing 1/k for k from 1 to n. For large n, the sum can be approximated by ln(n) + γ, where γ is the Euler-Mascheroni constant (approximately 0.57721). The calculator provides both exact (as a fraction) and decimal results, useful for mathematical and physical analyses.
Use this calculator in algorithm analysis (e.g., complexity of sorting algorithms), statistics (Zipf distribution), or physics (Fourier series). It is also helpful for students who want to verify manual calculations or explore the behavior of the harmonic series for different n values.
Caution: the harmonic series diverges, so for very large n the sum may exceed computational limits when represented as a fraction. Also, the logarithmic approximation is only accurate for large n; for small n, use the exact calculation. Remember that Hₙ is not defined for non-integer or negative n.
Frequently asked questions
What is the Euler-Mascheroni constant?
It is the limit of the difference between the harmonic sum Hₙ and the natural logarithm ln(n), approximately 0.57721. It appears in many areas of mathematics.
Does the harmonic series converge?
No, it diverges slowly to infinity. Although the terms decrease, the total sum grows without bound as n approaches infinity.
Can I use non-integer or negative n?
No. The harmonic sum is defined only for positive integers. For non-integer values, there are generalizations like the harmonic function, but not in this calculator.
What is the difference between exact and decimal result?
The exact result is a fraction representing the precise sum, useful for theoretical calculations. The decimal is an approximation, more practical for applications.
Why is the harmonic series important?
It appears in optimization problems, number theory, algorithm analysis, and physics. Studying it helps understand divergent series and approximations.