The irrationality of e can be proved using the infinite series expansion.
Assume that e is rational.i.e where and .
Since e is positive and 2.5<e<3(I’m not going to prove this.It’s quite simple.,use this as a lower bound for e and proceed to the upper bound)
This ensures that ie
Multiplying both sides by ;
and the sum are integers.So the infinite sum must also be an integer.
Now,we proceed to finding an upper bound or A,closing in on our contradiction.
The infinite sum which provides the upper bound can be maximized by minimizing the value of q.Taking the minimum value of q which is 3.
Since A is clearly non-negative,this is a contradiction as A must be an integer.