Hölder summation

Template:Short description In mathematics, Hölder summation is a method for summing divergent series introduced by Template:Harvs.

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

$\underset{n\to \mathrm{\infty }}{\lim }{H}_n^k$

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:

$\underset{\begin{array}{c}n\to \mathrm{\infty }\\ k\to \mathrm{\infty }\end{array}}{\lim }{H}_n^k$

See also

• Cesàro summation

References

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