# Difference between revisions of "Hölder summation methods"

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$$H_n^0=s_n=\sum_{k=0}^na_k,$$ | $$H_n^0=s_n=\sum_{k=0}^na_k,$$ | ||

− | $$H_n^k=\frac{H_0^{k-1}+\ | + | $$H_n^k=\frac{H_0^{k-1}+\dotsb+H_n^{k-1}}{n+1},$$ |

− | $k=1,2,\ | + | $k=1,2,\dotsc$. In particular, $(H,0)$-summability of a series indicates that it converges in the ordinary sense; $(H,1)$ is the method of arithmetical averages. The $(H,k)$-methods are totally [[Regular summation methods|regular summation methods]] for any $k$ and are compatible for all $k$ (cf. [[Compatibility of summation methods|Compatibility of summation methods]]). The power of the method increases with increasing $k$: If a series is summable to a sum $s$ by the method $(H,k)$, it will also be summable to that sum by the method $(H,k')$ for any $k'>k$. For any $k$ the method $(H,k)$ is equipotent and compatible with the Cesàro summation method of the same order $k$ (cf. [[Cesàro summation methods|Cesàro summation methods]]). If a series is summable by the method $(H,k)$, its terms $a_n$ necessarily satisfy the condition $a_n=o(n^k)$. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Hölder, "Grenzwerthe von Reihen an der Konvergenzgrenze" ''Math. Ann.'' , '''20''' (1882) pp. 535–549</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.H. Hardy, "Divergent series" , Oxford Univ. Press (1949)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Hölder, "Grenzwerthe von Reihen an der Konvergenzgrenze" ''Math. Ann.'' , '''20''' (1882) pp. 535–549</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.H. Hardy, "Divergent series" , Oxford Univ. Press (1949)</TD></TR></table> |

## Latest revision as of 13:46, 14 February 2020

A collection of methods for summing series of numbers, introduced by O. Hölder [1] as a generalization of the summation method of arithmetical averages (cf. Arithmetical averages, summation method of). The series

$$\sum_{n=0}^\infty a_n$$

is summable by the Hölder method $(H,k)$ to sum $s$ if

$$\lim_{n\to\infty}H_n^k=s,$$

where

$$H_n^0=s_n=\sum_{k=0}^na_k,$$

$$H_n^k=\frac{H_0^{k-1}+\dotsb+H_n^{k-1}}{n+1},$$

$k=1,2,\dotsc$. In particular, $(H,0)$-summability of a series indicates that it converges in the ordinary sense; $(H,1)$ is the method of arithmetical averages. The $(H,k)$-methods are totally regular summation methods for any $k$ and are compatible for all $k$ (cf. Compatibility of summation methods). The power of the method increases with increasing $k$: If a series is summable to a sum $s$ by the method $(H,k)$, it will also be summable to that sum by the method $(H,k')$ for any $k'>k$. For any $k$ the method $(H,k)$ is equipotent and compatible with the Cesàro summation method of the same order $k$ (cf. Cesàro summation methods). If a series is summable by the method $(H,k)$, its terms $a_n$ necessarily satisfy the condition $a_n=o(n^k)$.

#### References

[1] | O. Hölder, "Grenzwerthe von Reihen an der Konvergenzgrenze" Math. Ann. , 20 (1882) pp. 535–549 |

[2] | G.H. Hardy, "Divergent series" , Oxford Univ. Press (1949) |

**How to Cite This Entry:**

Hölder summation methods.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=H%C3%B6lder_summation_methods&oldid=32581