Difference between revisions of "Arithmetical averages, summation method of"
From Encyclopedia of Mathematics
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One of the methods for summing series and sequences. The series | One of the methods for summing series and sequences. The series | ||
| − | + | $$\sum_{k=0}^\infty u_k.$$ | |
is summable by the method of arithmetical averages to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013400/a0134002.png" /> if | is summable by the method of arithmetical averages to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013400/a0134002.png" /> if | ||
| − | + | $$\lim_{n\to\infty}\frac{s_0+\ldots+s_n}{n+1}=s,$$ | |
| − | where | + | where $s_n=\sum\nolimits_{k=1}^nu_k$. In this case, one also says that the sequence $\{s_n\}$ is summable by the method of arithmetical averages to the limit $s$. The summation method of arithmetical averages is also called the Cesàro summation method of the first order (cf. [[Cesàro summation methods|Cesàro summation methods]]). The summation method of arithmetical averages is completely regular (see [[Regular summation methods|Regular summation methods]]) and translative (see [[Translativity of a summation method|Translativity of a summation method]]). |
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | Instead of "arithmetical averages" the term "summation method of arithmetical | + | Instead of "arithmetical averages" the term "summation method of arithmetical means" is sometimes used, cf. [[#References|[a1]]], and instead of "summation" one also uses "summability" : summability method. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)</TD></TR></table> | ||
Revision as of 15:48, 9 April 2014
One of the methods for summing series and sequences. The series
$$\sum_{k=0}^\infty u_k.$$
is summable by the method of arithmetical averages to the sum 
if
$$\lim_{n\to\infty}\frac{s_0+\ldots+s_n}{n+1}=s,$$
where $s_n=\sum\nolimits_{k=1}^nu_k$. In this case, one also says that the sequence $\{s_n\}$ is summable by the method of arithmetical averages to the limit $s$. The summation method of arithmetical averages is also called the Cesàro summation method of the first order (cf. Cesàro summation methods). The summation method of arithmetical averages is completely regular (see Regular summation methods) and translative (see Translativity of a summation method).
References
| [1] | G.H. Hardy, "Divergent series" , Clarendon Press (1949) |
Comments
Instead of "arithmetical averages" the term "summation method of arithmetical means" is sometimes used, cf. [a1], and instead of "summation" one also uses "summability" : summability method.
References
| [a1] | R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950) |
How to Cite This Entry:
Arithmetical averages, summation method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetical_averages,_summation_method_of&oldid=31461
Arithmetical averages, summation method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetical_averages,_summation_method_of&oldid=31461
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article