Harmonic Number Calculator Nth Term

Harmonic Number Formula:

\[ H_n = \sum_{k=1}^n \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \]

Harmonic Number (H_n):

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1. What is the Harmonic Number?

The harmonic number H_n is the sum of the reciprocals of the first n natural numbers. It appears in many areas of mathematics, particularly in number theory and analysis.

2. How Does the Calculator Work?

The calculator uses the harmonic series formula:

\[ H_n = \sum_{k=1}^n \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \]

Where:

\( H_n \) — nth harmonic number (unitless)
\( n \) — positive integer term number

Explanation: The calculator sums the reciprocals of all integers from 1 to n to compute the nth harmonic number.

3. Importance of Harmonic Numbers

Details: Harmonic numbers have applications in algorithm analysis (especially quicksort), physics (such as in the analysis of the harmonic oscillator), and appear in various mathematical series and integrals.

4. Using the Calculator

Tips: Enter a positive integer n to calculate the nth harmonic number. The result is unitless and typically displayed with 6 decimal places.

5. Frequently Asked Questions (FAQ)

Q1: What's the behavior of harmonic numbers as n increases?
A: Harmonic numbers grow logarithmically. H_n ≈ ln(n) + γ + 1/(2n), where γ is the Euler-Mascheroni constant (~0.5772).

Q2: Are harmonic numbers ever integers?
A: No, except for H₁ = 1, all harmonic numbers are non-integers.

Q3: What's the relationship with the harmonic series?
A: The harmonic series is the infinite series whose partial sums are the harmonic numbers.

Q4: Are there generalizations of harmonic numbers?
A: Yes, generalized harmonic numbers include weights for each term, like H_n^(m) = Σ 1/k^m.

Q5: How accurate is this calculator?
A: It provides results accurate to 6 decimal places for reasonable values of n (up to about 10⁶).