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A summation method (cf. [[Summation methods|Summation methods]]) having the properties of linearity:
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A summation method (cf. [[Summation methods]]) having the properties of linearity:
  
1) if the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594601.png" /> is summable by the summation method to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594602.png" />, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594603.png" /> is summable by this method to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594604.png" />;
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594606.png" /> are summable by the summation method to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594608.png" /> respectively, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l0594609.png" /> is summable by this method to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l05946010.png" />.
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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|matrix summation method]] and a [[Semi-continuous summation method|semi-continuous summation method]]. There are non-linear summation methods. For example, the method in which summability of a series to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l05946011.png" /> is defined by the existence of the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l05946012.png" /> of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l05946013.png" />, where
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l05946014.png" /></td> </tr></table>
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$$T_n=\frac{s_{n+1}s_{n-1}-s_n^2}{s_{n+1}+s_{n-1}-2s_n}$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059460/l05946015.png" /> are the partial sums of the series), is not linear.
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($s_n$ are the partial sums of the series), is not linear.
  
 
====References====
 
====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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR>
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<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>
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<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>
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</table>
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[[Category:Sequences, series, summability]]

Latest revision as of 20:05, 9 November 2014

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=17658
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article