This is the third part in a 3-part blog post in which we prove that is transcendental.
Recall that in step 1 and step 2 we proved that for any prime sufficiently large and that is a nonzero integer. In this step we will prove that if is large enough, then and hence cannot be a nonzero integer. This will give a contradiction and we will conclude that is transcendental.
Recall that . Expanding this using our polynomial
Recall that and . So , , and . So we have
and (using the well-known fact that in the limit factorials grow faster than exponentials)
In particular, we can choose the prime large enough so that for , . So
as promised. This yields a contradiction and we conclude that is transcendental.