The number e is readily calculated to as many digits as desired using the rapidly converging series
e = 1 + 1 + 1 / 2! + 1 / 3! + 1 / 4! + ...
This series can also be used to show that e is irrational. Do so by completing the following argument. Suppose that e = p/q, where p and q are positive integers. Choose n > q and let
M = n! (e - 1 - 1 - 1 / 2! - 1 / 3! - ... - 1 / n!)
Now M is a positive integer. (Why?) Also,
M = n! [1 / (n + 1)! + 1 / (n + 2)! + 1 / (n + 3)! + ...]
= 1 / n + 1 + 1 / (n + 1) (n + 2) + 1 / (n + 1) (n + 2) (n + 3) + ....
< 1 / 1 + 1 + 1 / (n + 1)2 + 1 / (n + 1)3 + ...
= 1 / n
Which gives a contradiction (to what?)