Power multiplier in the summation of trigonometric series

Abstract

In this article we review the methods of power summation of factors. The degree of factors which
are arbitrary powers of summation indices are considered. We show that using the Poisson-Abel
method only those series can be summarized the order of member increase of which is
proportional to the exponent depending on the summation index. At the same time the Gauss-
Weierstrass method and other power factors methods can be also applied to the series the terms of
which increase in proportion to the exponential dependence of the indices summation.