1. What is the Harmonic Series?

The harmonic series is the infinite series \( H = \sum_{k=1}^{\infty} \frac{1}{k} \). Although it diverges to infinity, its partial sums \( H_n = \sum_{k=1}^{n} \frac{1}{k} \) grow logarithmically and are important in many areas of mathematics.

2. How Does the Calculator Work?

The calculator computes both the exact sum and the logarithmic approximation:

Where:

• \( H_n \) — Partial sum of harmonic series
• \( n \) — Number of terms
• \( \gamma \) — Euler-Mascheroni constant (≈0.57721)

Explanation: The approximation becomes better as n increases, with the difference \( H_n - \ln(n) \) approaching γ as n → ∞.

3. Importance of Harmonic Series

Details: The harmonic series appears in analysis, number theory, physics, and engineering. It's fundamental in understanding series convergence and appears in algorithms analysis.

4. Using the Calculator

Tips: Enter any positive integer n to calculate the partial sum H_n and its logarithmic approximation.

5. Frequently Asked Questions (FAQ)

Q1: Why does the harmonic series diverge?
A: Although the terms approach zero, they do so slowly enough that the sum grows without bound.

Q2: How accurate is the approximation?
A: The approximation improves with larger n. For n=100, the error is about 0.00995.

Q3: What is the exact value of γ?
A: γ is an irrational number (≈0.5772156649) defined as the limit of \( H_n - \ln(n) \) as n → ∞.

Q4: Are there faster ways to compute H_n?
A: For large n, using the approximation is much faster than summing terms directly.

Q5: Where is the harmonic series used?
A: It appears in analysis of algorithms, physics (e.g., pendulum periods), and number theory.