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.

Particularly, since the Cesàro sum of a convergent series always exists, the Hölder sum of a series (that is Hölder summable) can be written in the following form:

${\displaystyle \lim _{\begin{smallmatrix}n\rightarrow \infty \\k\rightarrow \infty \end{smallmatrix}}H_{n}^{k}}$

See also

• Cesàro summation

References

• Hölder, O. (1882), "Grenzwerthe von Reihen an der Konvergenzgrenze", Math. Ann., 20 (4): 535–549, doi:10.1007/bf01540142, S2CID 124308783
• "Hölder summation methods", Encyclopedia of Mathematics, EMS Press, 2001 [1994]