Check my proof and see if there are any mistakes or needs adjustment.

Given

If

then

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.