Free Harmonic Number Calculator

The harmonic number calculator (also referred to as a harmonic series calculator or Hn calculator) provides a straightforward way to compute the nth partial sum of the harmonic series. Whether you need the exact fraction for an integer input or a decimal approximation for a non‑integer value, this free online tool delivers the result quickly. It is equally useful for finding the sum of the harmonic series up to a given term, thus serving as an nth partial sum calculator as well.

Definition of a Harmonic Number

For a natural number $n$, the $n$‑th harmonic number $H_n$ is defined as the sum of the reciprocals of the first $n$ positive integers:

$$H_n = \sum_{k=1}^{n} \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}.$$

These numbers increase very slowly; for instance, $H_{1000} \approx 7.485$. They are closely related to the natural logarithm via the approximation $H_n \approx \ln n + \gamma$, where $\gamma \approx 0.5772156649$ is the Euler‑Mascheroni constant. The connection is made precise by the integral $\ln n = \int_{1}^{n} \frac{1}{x}\,dx$, which also explains the logarithmic growth. A well‑known result is that $H_n$ is never an integer for $n > 1$.

Calculating Hₙ for Integer n

For an integer $n$, $H_n$ can be obtained manually by adding the reciprocals of $1, 2, \dots, n$. The table below shows the first few harmonic numbers in both fractional and decimal forms:

$n$	$H_n$ (fraction)	$H_n$ (decimal)
1	1	1.00000
2	3/2	1.50000
3	11/6	1.83333
4	25/12	2.08333
5	137/60	2.28333
8	761/280	2.71786

The calculator displays the rational form (reduced to lowest terms) alongside the decimal when $n$ is an integer. This makes it simple to verify results or use the exact fraction in further calculations.

Extension to Non‑Integer Values

Although the elementary definition applies only to natural numbers, the digamma function $\psi(z)$ allows an analytic continuation of $H_n$ to positive real (and complex) arguments. The key relation is:

$$H_n = \psi(n+1) + \gamma,$$

where $\psi(z) = \Gamma'(z)/\Gamma(z)$ and $\gamma$ is the Euler‑Mascheroni constant. For a positive non‑integer $n$, the digamma function can be evaluated efficiently using the series:

$$\psi(z+1) = -\gamma + \sum_{k=1}^{\infty} \frac{z}{k(k+z)},$$

which converges for all $z$ with positive real part. The calculator automates this procedure, so the user never has to handle the digamma function directly. The result is returned as a decimal rounded to five decimal places.

Relation to the Infinite Harmonic Series

The infinite harmonic series $\sum_{k=1}^{\infty} \frac{1}{k}$ diverges—its partial sums grow without bound. The harmonic number $H_n$ is precisely the $n$‑th partial sum of this series. Therefore, the harmonic series calculator can be used to determine the sum of the harmonic series for any finite number of terms, serving as a sum‑of‑harmonic‑series calculator.

Approximate Formula

For quick estimates, the Euler–Maclaurin expansion provides an excellent approximation:

$$H_n \approx \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^{2}} + \frac{1}{120n^{4}} - \cdots.$$

Even using only the first three terms yields an error smaller than $0.001$ for $n \ge 10$.

How to Use the Calculator

1. Enter a positive number $n$ (integer or non‑integer) in the input field.
2. The tool instantly computes the corresponding harmonic number $H_n$.
3. For integer $n$, the result is shown both as a simplified fraction and as a decimal with five‑digit precision.
4. For non‑integer $n$, only the decimal approximation is displayed.
5. To ensure fast response, the maximum allowed integer $n$ is $10^{6}$; for non‑integer $n$ the limit is $10^{3}$. Values beyond these limits can be handled upon request.

This harmonic number calculator is designed to be intuitive and accessible, making it a reliable choice for anyone who needs to evaluate harmonic numbers or explore the properties of the harmonic series.