Check my proof and see if there are any mistakes or needs adjustment.
If then consider the function this function is positive and countinous for . Also . By the first derivative test the function is increasing for and decreasing for . Thus is a positive, decreasing and countinous function for
The gamma function can be expressed as
The first integral definitely converges so the convergence of the gamma function determines on the improper integral . By the integral test for infinite series this improper integral converges if and only if converges. But by the ratio test Thus, the improper integral converges, thus, converges for .
Now finally, if then . But then with converges just as was proven in the last paragraph. Thus, the Gamma function converges for all
Note: is the fundamental property of the Gamma function and was assumed to have been known.
Wow, I finally finished, you have no idea how long it took me to write this in Latex form.