# Matrix summation method

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One of the methods for summing series and sequences using an infinite matrix. Employing an infinite matrix , a given sequence is transformed into the sequence : If the series on the right-hand side converges for all and if the sequence has a limit for : then the sequence is said to be summable by the method determined by the matrix , or simply summable by the matrix , and the number is referred to as its limit in the sense of this summation method. If is regarded as the sequence of partial sums of a series (1)

then this series is said to be summable to the sum by the matrix .

A matrix summation method for series can be also defined directly by transforming the series (1) into a sequence : (2)

where is a given matrix. In this case the series (1) is said to be summable to the sum if, for all the series on the right-hand side in (2) converges and Less often used are matrix summation methods defined by a transformation of a series (1) into a series (3)

where or by a transformation of a sequence into a series (4)

where which use matrices and , respectively. In these cases the series (1) with the partial sums is summable to the sum if the series (3) converges to or, respectively, if the series (4) converges to .

The matrix of a summation method all entries of which are non-negative is called a positive matrix. Among the matrix summation methods one finds, for example, the Voronoi summation method, the Cesàro summation methods, the Euler summation method, the Riesz summation method , the Hausdorff summation method, and others (see also Summation methods).

How to Cite This Entry:
Matrix summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_summation_method&oldid=12057
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article