# Convergence of the gamma function

I will write up the proof of the convergence of the gamma function as a follow-up to this post, where I show why the exponential grows faster than any polynomial. The gamma function is defined by

**Proposition:** This integral converges for

**Proof:** Let’s divide the integral in a sum of two terms,

For the first term, since the function is decreasing, it’s maximum on the interval is attained at so

But for this last integral converges to

For the second term, we use what we showed in this post: since the exponential grows faster than any polynomial, for every we can take so big that So

which completes the proof.

May I ask why e^(t/2) was chosen not e^(t/3) for instance for the second term? Much appreciated.

I don’t think it makes a difference what is used (e^(t/2) or something else) so long as it is in fact > x^(p-1). For instance what if we used e^x? e^x > x^(p-1). Using e^x would make it even more obvious.

It is true that we could also use because it’s greater than for any , but also because is integrable. What you suggest Madeline doesn’t work, if we take instead of , then we get and we can’t conclude anything.