Motivation of irrationality measure

Question 1

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 $e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^n}{n!}+ \cdots$ So the series for $e^{-1}$ is $s_n = \sum_{k=0}^{n} \frac{(-1)^{k}}{k!}$ so that $0 < e^{-1}-s_{2k-1} < \frac{1}{(2k)!}$ or $0 < (2k-1)!(e^{-1}-s_{2k-1}) < \frac{1}{2k} \leq \frac{1}{2}$ If $e^{-1}$ were rational then we would have a difference of two integers which is not an integer (i.e. between $0$ and $\frac{1}{2}$ ).

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 $\pi$ ?

Question 2

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.

Motivation of irrationality measure

Answer

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