Harmonic Number Calculator

This calculator computes the ﻿nth​
 harmonic number, which is the sum of the reciprocals of the positive integers up to integer ﻿n
. If the ﻿n
 is non-integer, the calculator computes the approximate harmonic number by using the digamma function and the Euler–Mascheroni constant.
CalculationInputs﻿﻿n
:5.00​
﻿
Output﻿﻿Hn
:2.28​
﻿
Integer ﻿n
﻿
﻿﻿Hn​​=∑k=1n​​k1​​
﻿
Non-Integer ﻿n
﻿
﻿﻿Hn​​≈ψ(n+1)+γ
﻿
Explanation The harmonic series approximates the natural logarithm plus the Euler-Mascheroni constant, that is:
﻿
Hn≈ln(n)+γH_n \approx ln(n)+\gammaHn​≈ln(n)+γ
This approximation becomes more accurate as ﻿n
 becomes larger. Because the harmonic series approximates the natural logarithm, it is a divergent series which means it grows infinitely.
﻿
Graph showing how the harmonic series is approximates the logarithmic function
The harmonic number is calculated in two different ways for integer and non-integer values of ﻿n
: 
Integer ﻿n
﻿
﻿﻿Hn​​=∑k=1n​​k1​​
﻿
Non-Integer ﻿n
﻿
﻿﻿Hn​​≈ψ(n+1)+γ
﻿
Where:
﻿﻿n
 is the term at which the calculation of the harmonic number (also known as the series sum) ends. The input can be an integer or a non-integer.
﻿﻿Hn​
 is the ﻿nth​
 harmonic number. For integers, it's the sum of the reciprocals of the positive integers up to integer ﻿n
. For non-integers, it's approximated using the digamma function and the Euler–Mascheroni constant.
﻿﻿ψ(n+1)
 is the digamma function, which is the logarithmic derivative of the gamma function, given by ﻿ψ(x)=dxd​​ln[Γ(x)]
. It is used to approximate ﻿Hn​
 for non-integer values of ﻿n
.
﻿﻿Γ(x)
 is the gamma function, which extends the factorial function to complex and real number arguments, except for negative integers. It is defined for positive real numbers ﻿x
 as: ﻿Γ(x)=∫0∞​​tx−1e−tdt
. The gamma function is related to factorials by ﻿Γ(n)=(n−1)!
 for any natural number ﻿n
.
﻿﻿γ
 is the Euler–Mascheroni constant, which is a mathematical constant approximately equal to﻿γ
:0.5772156649​
﻿
﻿
This calculator is particularly useful in scenarios such as:
Mathematics: Harmonic numbers play a key role in analyzing series convergence and divergence.
Computer Science: Harmonic numbers are crucial for algorithm analysis, particularly in sorting and hashing, by estimating average case scenarios.
Physics: Harmonic numbers explain overtones in music and resonances in circuits, key to understanding wave patterns and vibrations.
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