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Harmonic Numbers

Definition

H_n = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{n} = \sum_{k=1}^{n} \frac{1}{k}, n \geq 0 \tag{1}

Lower Bound

H_{2^m} \geq 1 + \frac{m}{2} \tag{2}
\sum_{k=1}^{n} H_k = (n + 1)H_n - n \tag{3}

Using Euler's Constant to approximate size of H_n

H_n = \ln{n} + \gamma + \frac{1}{2n} - \frac{1}{12n} + \frac{1}{120n^4} - \epsilon, 0 < \epsilon < \frac{1}{252n^6}, \gamma = 0.5772156649 \tag{4}

Where \gamma is Euler's constant.

Last update: March 31, 2022