A New Test Method on the Convergence and Divergence for Infinite Integral

—The way to distinguish convergence or divergence of an infinite integral on non-negative continuous function is the important and difficult question in the mathematical teaching all the time. Using the comparison of integrands to judge some exceptional infinite integrals are hard or even useless. In this paper, we establish the exponential integrating factor of negative function, and present a new method to test based on its exponential integrating factor. These conclusions are convenient and valid additions of the previously known results.


I. INTRODUCTION
The convergence and divergence of infinite integral plays a significant role in mathematical analysis, and has been received much attention of many researchers.Many effective methods have been proposed such as Ordinary Comparison Test, Limiting Test, Dirichlet Test and Abel Test [1].The basic ideas of these methods are to find another comparison infinite integral that its convergence or divergence is certain.But it is difficult or impossible to find the comparison infinite.So these arouse great interest of many scholars to research ( [2][3]).
It is well-known that the convergence and divergence of infinite integral for the different integrand which its limit is zero is different.Base on the geometric meaning of infinite integral, it equal to the size of the area which is surrounded by the integrand (image) and X axis (not closed).The Convergence and divergence mainly determined by the proximity degree of integrand tends to the x-axis.In this paper, we discuss the test method for the convergence and divergence of infinite integral () the problem by ln ( ) / ln f x x [3], it can't intuitive describe the proximity degree of integrand tends to the x-axis, so the exponential integrating factor of negative function is created and we obtain a new test method for the convergence and divergence of infinite integral.This method is more simple and feasible than ever before.( ) / ( ) f x g x is finally monotone decreasing, then (i) 00 , , ( ) ( ) Proof: Test, we obtain this conclusion.
Assume that we know the convergence and divergence of an infinite integral, we can obtain the convergence and divergence of other infinite integrals by comparing the size of the exponential factor.But it's complex.In fact, it can be derived alone by the exponential factor () Combining [3], we have that () possible examples.In fact, we can continue to study convergence as follow: a  and lim ( ) Proof: (i) We can obtain that () x R x q using theorem 2.4, we get that the integration is convergent when 1 q   the integration is divergent when 1 q   Example2: Discuss the convergence of Using the exponential integrating factor of ( ) ( ) In fact, we can construct some convergent infinite integral for some specific needs.(ii) Easy to get by (i). (iii

IV. CONCLUSIONS AND PROSPECT
In this paper, we get a new test method for the convergence and divergence of infinite integral, as everyone knows, defect integral can be changed into infinite integral by 1 x t  so we can expands our method on the defect integral.
We can almost judge the convergence of the infinite integrals by the conclusions refer to in Section 2. So we discuss some questions of infinite integral by the properties of the convergent infinite integral.In fact, the most important application in Section 3 is to construct plenty of distributions in the probability field especially in the Risk Theory.The claim distribution is vital for both the insured and the insurance company [4][5][6][7], constructing suitable is beneficial to need for the insured or development of the insurance companies.This is a topic for future research.
give the following conditions of the nonnegative function () gx According to the Ordinary