View source for 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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  S.A. Baron,   "Introduction to the theory of summability of series" , Tartu  (1966)  (In Russian)</TD></TR>
</table>

[[Category:Sequences, series, summability]]