Suppose e^2 is rational, then (e^2 - 1)/(e^2 + 1) = (e - 1/e)/(e + 1/e) is also rational. Let it be p/q , where p and q are positive integers and q>p and q>1. Expanding (e - 1/e)/(e + 1/e), we have:

(1/1! + 1/3! + 1/5! +...)/(1 + 1/2! + 1/4! +...) = p/q

multiply q! up and down on the left :

q!*(1/1! + 1/3! + 1/5! +...)/q!*(1 + 1/2! + 1/4! +...) = p/q

or:

(q!*a + q!*b)/(q!*x + q!*y) = p/q (1)

where

a=1/1! + 1/3! + 1/5! +...+1/r!, (r <= q)
b=1/(r+2)! + 1/(r+4)! + ...

x=1 + 1/2! + 1/4! +...+1/s!, (s <= q-1)
y=1/(s+2)! + 1/(s+4)! + ...

rearrange (1):

q!*a - p*(q-1)!*x = p*(q-1)!*y - q!*b (2)

The left can be proven to be non-zero integer;
It is easy to prove that the right of (2) < 1, contrdicting the left.

Therefore e^2 is iirational.