Proof that e is irrational

The series expansion of the number e can be used to prove that e is irrational.

Suppose that e = a/b, for some positive integers a and b. If we multiply each side of the series expansion

by b!, we obtain

The first term on the right side of this equation is an integer. The remainder of the right side is a positive number bounded above by the geometric series

Since b > 1, this means the entire right side of the original equation cannot be an integer. But this a contradiction, for b!e = a(b-1)! is clearly an integer. This completes the proof.

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