Proof that e is irrational
The series expansion
-
of the number e can be used to prove that e is irrational.
Suppose e = a/b, for some positive integers a and b. Consider the number
We will show that x is a positive integer less than 1, and this contradiction will establish the irrationality of e.
- To see that x is an integer, note that
Here, the last term in the final sum is to be interpreted as an empty product.
- To see that x is a positive number less than 1, note that
Here, the last sum is a geometric series.
Since there does not exist a positive integer less than 1, we have reached a contradiction, and so e must be irrational. This completes the proof.
Q.E.D
