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 + ⋯ , {\displaystyle a_{1}+a_{2}+\cdots ,}

define

H n 0 = a 1 + a 2 + ⋯ + a n {\displaystyle H_{n}^{0}=a_{1}+a_{2}+\cdots +a_{n}}

H n k + 1 = H 1 k + ⋯ + H n k n {\displaystyle H_{n}^{k+1}={\frac {H_{1}^{k}+\cdots +H_{n}^{k}}{n}}}

If the limit

lim n → ∞ H n k {\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:

lim n → ∞ k → ∞ H n k {\displaystyle \lim _{\begin{smallmatrix}n\rightarrow \infty \\k\rightarrow \infty \end{smallmatrix}}H_{n}^{k}}

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