# Linear summation method

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

1) if the series $\sum_{k=0}^\infty a_k$ is summable by the summation method to the sum $A$, then the series $\sum_{k=0}^\infty ca_k$ is summable by this method to the sum $cA$;

2) if the series $\sum_{k=0}^\infty a_k$, $\sum_{k=0}^\infty b_k$ are summable by the summation method to $A$ and $B$ respectively, then the series $\sum_{k=0}^\infty(a_k+b_k)$ is summable by this method to the sum $A+B$.

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 $S$ is defined by the existence of the limit $S$ of the sequence $\{T_n\}$, where

$$T_n=\frac{s_{n+1}s_{n-1}-s_n^2}{s_{n+1}+s_{n-1}-2s_n}$$

($s_n$ 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)
How to Cite This Entry:
Linear summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Linear_summation_method&oldid=34458
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article