Harmonic Number Approximation Formula
Harmonic Number Approximation Formula
\[H(n)=\ln(n)+\gamma+\frac{1}{2n}\]
Variables
H = harmonic number approximation
n = index value
γ = 0.5772156649
Description
What is this formula?
This formula approximates the harmonic number using a logarithmic expression and Euler–Mascheroni constant.
When to use it
Use it when estimating harmonic sums for large values of n without computing full summation.
Example
Data:
n = 10
Formula:
H[n]=log(n)+0.5772156649+1/(2*n)
Substitution:
H[10]=log(10)+0.5772156649+0.05
Result:
H[10] ≈ 3.03
Applications
Used in number theory, computer science, and asymptotic analysis.
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