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:
hölder, summation, mathematics, method, summing, divergent, series, introduced, hölder, 1882, definition, editgiven, series, displaystyle, cdots, nbsp, define, displaystyle, cdots, nbsp, displaystyle, frac, cdots, nbsp, limit, displaystyle, rightarrow, infty, . In mathematics Holder summation is a method for summing divergent series introduced by Holder 1882 Definition editGiven a series a 1 a 2 displaystyle a 1 a 2 cdots nbsp define H n 0 a 1 a 2 a n displaystyle H n 0 a 1 a 2 cdots a n nbsp 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 nbsp If the limit lim n H n k displaystyle lim n rightarrow infty H n k nbsp exists for some k this is called the Holder sum or the H k sum of the series Particularly since the Cesaro sum of a convergent series always exists the Holder sum of a series that is Holder 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 nbsp See also editCesaro summationReferences editHolder O 1882 Grenzwerthe von Reihen an der Konvergenzgrenze Math Ann 20 4 535 549 doi 10 1007 bf01540142 S2CID 124308783 Holder summation methods Encyclopedia of Mathematics EMS Press 2001 1994 nbsp This mathematical analysis related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Holder summation amp oldid 1161264749, wikipedia, wiki, book, books, library,