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

$H^0_n=a_1+a_2+\cdots+a_n \,$

$H^{k+1}_n=\frac{H^k_1+\cdots+H^k_n}{n}$

If the limit

$\lim_{n\rightarrow\infty}H^k_n \,$

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