The present paper investigates the absolutely convergence for Fourier-Laplace series of the functions belonging to some smoothing functions class given on the unit sphere and defined k-th order difference with step t along a geodesic emanating from μ∈Ω n,averaged ov.

A strict proof of the formulas of the convergence real part,absolutely convergence part and uniform convergence real part for the exponential series has been given.

Making use of the result, we can directly distinguish whether alternate series are convergence or not, absolutely convergence or conditioned convergence.

Subseries convergence and absolute convergence in locally convex spaces;

The concept of absolute convergence is generalized.

Using the comparison test,ratio test,or root value test,and their limit forms to judge the convergence and divergence of ∑∞n=1|un|,we can get if the series ∑∞n=1un is absolute convergence.

Some valuable properties of the set of the sum of absolute convergent series are obtained.

that in the definition ofexpectation condition that is absolutely convergent can be changed to the convergence of.

It is proved that p -normed vector space X is complete if and only if absolutely convergent series is convergent,and that in p -Banach spaces,abstract function s bounded variation is equivalent with its p -weakly bounded variation if and only if series s absolutely convergent is equivalent with its p -weakly absolufely convergent.