Proof that e is Irrational
The number e = 2.71828.. can be shown to be irrational by a very
simple argument based on the power series expansion of the exponential
function, which gives
1/e = 1/0! - 1/1! + 1/2! - 1/3! + 1/4! - ...
If P(k) is the kth partial sum, we see that P(k) - P(k-1) = +-1/k!,
and so k((k-1)!)P(k-1) - k!P(k) = +-1. It follows that placing
each pair of consecutive partial sums on a common basis, we have
2/ 6 < 1/e < 3/ 6
8/ 24 < 1/e < 9/ 24
44/120 < 1/e < 45/120
264/720 < 1/e < 265/720
and so on, where each pair of bounding numerators differs by 1, and
the denominators are m!. The first of these relations proves that
if 1/e is rational its denominator cannot be a divisor of 6, because
then it could be written n/6 for some integer n, and there is no
such integer greater than 2 and less than 3.
Similarly the next relation proves that the denominator of 1/e cannot
be a divisor of 24, and the next proves that it cannot be a divisor
of 120, and so on. Continuing in this way, it's clear that the
denominator of 1/e cannot be a divisor of any m! for m=2,3,4,...and
so on to infinity. But every integer k is a divisor of m! for all
m >= k, so 1/e (and therefore e) cannot be a rational number.
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