I have a question about the irrationality of e:
In proving the irrationality of e, one can prove the irrationality of e−1 by using the series ex=1+x+x22!+⋯+xnn!+⋯ So the series for e−1 is sn=n∑k=0(−1)kk! so that 0<e−1−s2k−1<1(2k)! or 0<(2k−1)!(e−1−s2k−1)<12k≤12 If e−1 were rational then we would have a difference of two integers which is not an integer (i.e. between 0 and 12).
Question: Is this "irrationality of e−1 by alternating series" proof what motivated the definition of irrationality measure? It seems that this was born out of using alternating series.
Does the same method work for π ?
The motivation for the definition of irrationality measure came from results in diophantine approximation. E.g., Dirichlet's Theorem states that for every real irrational x there are infinitely many integer pairs p,q with |x−(p/q)|<q−2. If x is a real quadratic irrational, then there's a positive constant C such that for every integer pair p,q we have |x−(p/q)|>Cq−2. Liouville's Theorem was mentioned in the first comment. These can all be stated in terms of irrationality measure, and form an obvious motivation for that concept.