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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.

See also harmonic series (mathematics).

Science Fair Project Encyclopedia Contents Page
10-26-2009 08:16:03
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