Hölder summation

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

Definition

Given a series

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

define

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

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

If the limit

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

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
• Hazewinkel, Michiel, ed. (2001), "Hölder summation methods", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4