In mathematics, Hölder summation is a method for summing divergent series introduced by Hölder (1882).

Definition

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Given a series

 

define

 
 

If the limit

 

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:

 

See also

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References

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  • 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]