Hölder summation

In mathematics, Hölder summation is a method for summing divergent series introduced by Hölder (1882).

Definition

Given a series

$a_{1}+a_{2}+\cdots ,\,$

define

$H_{n}^{0}=a_{1}+a_{2}+\cdots +a_{n}\,$

$H_{n}^{k+1}={\frac {H_{1}^{k}+\cdots +H_{n}^{k}}{n}}$

If the limit

$\lim _{n\rightarrow \infty }H_{n}^{k}\,$

exists for some k, this is called the Hölder sum, or the (H,k) sum, of the series.

References

• Hölder, O. (1882), "Grenzwerthe von Reihen an der Konvergenzgrenze", Math. Ann., 20: 535–549, doi:10.1007/bf01540142
• "Hölder summation methods", Encyclopedia of Mathematics, EMS Press, 2001 [1994]