# Difference between revisions of "Linear summation method"

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− | A summation method (cf. [[ | + | {{TEX|done}} |

+ | A summation method (cf. [[Summation methods]]) having the properties of linearity: | ||

− | 1) if the series | + | 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 | + | 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 [[ | + | 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==== | ====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> | + | <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]] |

## 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