Difference between revisions of "Arithmetical averages, summation method of"
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$$\sum_{k=0}^\infty u_k.$$ | $$\sum_{k=0}^\infty u_k.$$ | ||
| − | is summable by the method of arithmetical averages to the sum | + | is summable by the method of arithmetical averages to the sum $s$ if |
| − | $$\lim_{n\to\infty}\frac{s_0+\ | + | $$\lim_{n\to\infty}\frac{s_0+\dots+s_n}{n+1}=s,$$ |
where $s_n=\sum\nolimits_{k=1}^nu_k$. In this case, one also says that the sequence $\{s_n\}$ is summable by the method of arithmetical averages to the limit $s$. The summation method of arithmetical averages is also called the Cesàro summation method of the first order (cf. [[Cesàro summation methods|Cesàro summation methods]]). The summation method of arithmetical averages is completely regular (see [[Regular summation methods|Regular summation methods]]) and translative (see [[Translativity of a summation method|Translativity of a summation method]]). | where $s_n=\sum\nolimits_{k=1}^nu_k$. In this case, one also says that the sequence $\{s_n\}$ is summable by the method of arithmetical averages to the limit $s$. The summation method of arithmetical averages is also called the Cesàro summation method of the first order (cf. [[Cesàro summation methods|Cesàro summation methods]]). The summation method of arithmetical averages is completely regular (see [[Regular summation methods|Regular summation methods]]) and translative (see [[Translativity of a summation method|Translativity of a summation method]]). | ||
Latest revision as of 14:17, 30 December 2018
One of the methods for summing series and sequences. The series
$$\sum_{k=0}^\infty u_k.$$
is summable by the method of arithmetical averages to the sum $s$ if
$$\lim_{n\to\infty}\frac{s_0+\dots+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) |
Arithmetical averages, summation method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arithmetical_averages,_summation_method_of&oldid=31461