1. What is the Harmonic Series?

The harmonic series is the infinite series formed by summing the reciprocals of the positive integers. The n-th harmonic number is the partial sum of the first n terms of this series.

2. How Does the Calculator Work?

The calculator uses the harmonic number formula:

Where:

• \( H_n \) — n-th harmonic number
• \( n \) — Positive integer
• \( k \) — Index of summation

Explanation: The calculator computes the sum of reciprocals from 1 to n, providing the exact harmonic number for the given input.

3. Importance of Harmonic Numbers

Details: Harmonic numbers appear in many areas of mathematics including number theory, analysis, and combinatorics. They are important in the study of algorithms and appear in various approximation formulas.

4. Using the Calculator

Tips: Enter any positive integer value for n. The calculator will compute the sum of reciprocals from 1 to n. Larger values of n will take slightly longer to compute.

5. Frequently Asked Questions (FAQ)

Q1: Does the harmonic series converge?
A: The infinite harmonic series diverges (grows without bound), though very slowly. The partial sums grow approximately as ln(n) + γ where γ is the Euler-Mascheroni constant.

Q2: What are some applications of harmonic numbers?
A: They appear in analysis of algorithms (like Quicksort), physics (like the harmonic oscillator), and various probability problems.

Q3: Are there closed-form expressions for harmonic numbers?
A: For general n, there is no elementary closed form, though they can be expressed using digamma functions or as integrals.

Q4: What is the relationship to the Riemann zeta function?
A: The harmonic series is ζ(1), where ζ is the Riemann zeta function. Generalized harmonic numbers correspond to partial sums of ζ(s) for s > 1.

Q5: How quickly does H_n grow?
A: H_n grows logarithmically with n. For large n, H_n ≈ ln(n) + γ + 1/(2n), where γ ≈ 0.5772 is the Euler-Mascheroni constant.