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
Hazewinkel, Michiel, ed. (2001) [1994], "Hölder summation methods", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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