Improper integral with exponential function.

In this paper, we explore the improper integral with exponential function f = x^x is approached to infinite series, and also prove the convergence of these series. An improper integral converges if the limit defining it exists. We use Maple code to calculate the infinite series. The application of improper integral appear in several domain in science. As an application in this paper, three examples are given to illustrate the effectiveness of our main result. [ABSTRACT FROM AUTHOR]