Harmonic number

In mathematics, the generalized harmonic number of order $n$ is given by

$H^{(m)}_n=\sum_{k=1}^n \frac{1}{k^m}.$

The special case of $m = 1$ is simply called a harmonic number and is frequently written without the superscript, as

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

In the limit of $n\rightarrow \infty$ , the generalized harmonic number converges to the Riemann zeta function

$\lim_{n\rightarrow \infty} H^{(m)}_n = \zeta(m)$

The related sum $\sum_{k=1}^n k^m$ occurs in the study of Bernoulli numbers.

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