Harmonic number

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

$H_{n,m}=\sum _{k=1}^{n}{\frac {1}{k^{m}}}.$

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

$\lim _{n\rightarrow \infty }H_{n,m}=\zeta (m)$

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

Applications

The harmonic numbers appear in several calculation formulas, such as the digamma function:

$\psi (n)=H_{n-1}-\gamma \,$

where γ is the Euler-Mascheroni constant The harmonic numbers are also part of the definition of γ,

$\gamma =\lim _{n\rightarrow \infty }{\left(H_{n}-\ln(n)\right)}$

and may be calculated from the formula:

$H_{n}=\int _{0}^{1}{\frac {1-x^{n}}{1-x}}\,dx$

due to Euler

References

• How Euler Did It -- Estimating the Basel problem

See also

• harmonic series (mathematics).
• Riemann zeta function
• Digamma function

External links

• Harmonic Number -- from MathWorld