Harmonic Numbers

#Concrete Mathematics, Special Numbers

2022/09/06

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Table of Content

• Defintion
  • Harmonic numbers of order \(r\)
  • Riemann’s zeta function
• Identities
• Summation
• Reference

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Defintion

\[H_n = \sum_{k=1}^n {1 \over k},\ n \in \mathbb{N}\]

\(H_0 = \sum_{k=1}^0{1 \over k} = 0\)，沒有除以 \(0\) 的風險

Harmonic numbers of order \(r\)

\[H^{(r)}_n = \sum_{k=1}^n{1 \over k^r}\]

Riemann’s zeta function

\[\zeta(r) = H^{(r)}_\infty = \sum_{k \geq 1}{1 \over k^r}\]

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Identities

\[\lim_{n \to \infty}(H_n - \ln n) = \gamma \approx 0.577218\]

\(\gamma\): Euler’s constant

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Summation

\[\sum_{0 \leq k < n}H_k = nH_n - n \tag{1}\]

summation by parts

\[\sum_{k=1}^nH^{(2)}_k = (n+1)H^{(2)}_n - H_n \tag{2}\]

改變 sum 的順序

注意上界（含不含 \(n\)）

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Reference

• [CMath] (p.264), Concrete Mathematics: A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashni