Harmonic number

Task number: 3856

In the analysis of algorithms, the so-called harmonic number sometimes appears:\( H_n = 1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n} = \sum\limits_{k=1}^n \frac{1}{k}\).

  • Variant

    For \(n=2^m\) prove: \(1+\frac{m}{2} \leq H_n \leq 1+m\).

  • Variant

    Prove a more general estimate: \(\frac{1}{2}\log_2 n \leq H_n \leq 2+ \log_2 n\).

  • Variant

    Improve the general estimate onto: \(\ln n \leq H_n \leq 1+ \ln n\).

Difficulty level: Moderate task
Routine calculation training
Cs translation
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