This is the third part in a 3-part blog post in which we prove that is transcendental.

* Three-step proof that is transcendental*Step 1

Step 2

*Step 3*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.

** Step 3. **

Recall that . Expanding this using our polynomial

we obtain

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.

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