Linear summation method

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.

References

[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)