Approximate Size of Sum of Harmonic Series

Theorem

Let $H_n$ denote the sum of the harmonic series:

$H_n = \ds \sum_{k \mathop = 1}^n \frac 1 k$

Then $H_n$ can be approximated as follows:

$H_n \approx \ln n + \gamma + \dfrac 1 {2 n} - \dfrac 1 {12 n^2} + \dfrac 1 {120 n^4} - \epsilon$

where:

$\gamma$ denotes the Euler-Mascheroni constant: $\gamma \approx 0 \cdotp 57721 \, 56649 \, \ldots$

$0 < \epsilon < \dfrac 1 {252 n^6}$

Proof

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Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(3)$