View source for Regular summation methods

''permanent summation methods''

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Regular summation methods are those methods for summing series (sequences) that sum every convergent series (sequence) to the same sum as that to which it converges. Regular summation methods are a special case of conservative summation methods, which sum every convergent series (sequence) to a finite sum, although possibly different from that to which it converges. If a regular summation method is defined by the transformation of a sequence $(s_n)$ to a sequence $(\sigma_n)$ by means of an infinite matrix $[a_{nk}]$:
\begin{equation}
\label{eq1}
\sigma_n = \sum_{k=1}^{\infty} a_{nk}s_k, \quad n=1,2,\ldots
\end{equation}
(see [[Matrix summation method]]), then the transformation \ref{eq1} and the matrix of this transformation, $[a_{nk}]$, are called regular.

Many of the most common summation methods are regular. This applies to the [[Cesàro summation methods]] $(C,k)$ for $k \geq 0$, the [[Hölder summation methods]] and the [[Abel summation method]], among others. There are non-regular summation methods, such as the Cesàro summation method $(C,k)$ for $k<0$, and the [[Riemann summation method]].

A summation method is called completely regular if it is regular and if it sums every series (sequence) with real terms converging to $\infty$ (or $-\infty$) to $\infty$ (respectively, $-\infty$). A regular summation method defined by a positive matrix is completely regular (see also [[Regularity criteria]]).

====References==== 

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|valign="top"|{{Ref|Ba}}||valign="top"| S. Baron, "Introduction to theory of summation of series", Tallin (1977) (In Russian)
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|valign="top"|{{Ref|Co}}||valign="top"| R.G. Cooke, "Infinite matrices and sequence spaces", Macmillan (1950)
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|valign="top"|{{Ref|Ha}}||valign="top"| G.H. Hardy, "Divergent series", Clarendon Press (1949)
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|valign="top"|{{Ref|Ka}}||valign="top"| G.F. Kangro, "Theory of summability of sequences and series" ''J. Soviet Math.'', '''5''' : 1 (1976) pp. 1–45 ''Itogi Nauk. i Tekhn. Mat. Anal.'', '''12''' (1974) pp. 5–70
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