# Euler summation method

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One of the methods for summing series of numbers and functions. A series (*)

is summable by means of the Euler summation method ( -summable) to the sum if where and .

The method was first applied by L. Euler for to sum slowly-convergent or divergent series. Since the technique was later extended to arbitrary values of by K. Knopp , it is also known for arbitrary as the Euler–Knopp summation method. This method is regular for (see Regular summation methods); if a series is -summable, then it is also -summable, , to the same sum (see Inclusion of summation methods). For the summability of the series (*) by the Euler summation method implies that the series is convergent. If the series is -summable, then its terms satisfy the condition , . The Euler summation method can also be applied for analytic continuation beyond the disc of convergence of a function defined by means of a power series. Thus, the series is -summable to the sum in the disc with centre at and of radius .

How to Cite This Entry:
Euler summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_summation_method&oldid=19089
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article