Regular summation methods

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permanent summation methods

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 to a sequence by means of an infinite matrix :


(see Matrix summation method), then the transformation (*) and the matrix of this transformation, , are called regular.

Many of the most common summation methods are regular. This applies to the Cesàro summation methods for , the Hölder summation methods and the Abel summation method, among others. There are non-regular summation methods, such as the Cesàro summation method for , 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 (or ) to (respectively, ). A regular summation method defined by a positive matrix is completely regular (see also Regularity criteria).


[1] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[2] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[3] 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
[4] S. Baron, "Introduction to theory of summation of series" , Tallin (1977) (In Russian)
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Regular summation methods. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article