Linear summation method

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A summation method (cf. Summation methods) having the properties of linearity:

1) if the series is summable by the summation method to the sum , then the series is summable by this method to the sum ;

2) if the series , are summable by the summation method to and respectively, then the series is summable by this method to the sum .

All most widespread summation methods are linear; in particular, a matrix summation method and a semi-continuous summation method. There are non-linear summation methods. For example, the method in which summability of a series to the sum is defined by the existence of the limit of the sequence , where

( are the partial sums of the series), is not linear.


[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 (1976) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 5–70
[4] S.A. Baron, "Introduction to the theory of summability of series" , Tartu (1966) (In Russian)
How to Cite This Entry:
Linear summation method. 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