Hölder summation

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

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.

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:


 * $$\lim_{\begin{smallmatrix}

n\rightarrow\infty\\ k\rightarrow\infty \end{smallmatrix}}H^k_n $$