Following from the book "An Introduction to the Theory of Numbers" - Hardy & Wright I am having trouble with this proof. The book uses a familiar proof for the irrationality of e and continues into some generalizations that lose me.
In the following statement where is the series coming from or how is the statement derived?
$ f = f(x) = \frac{x^n(1 - x)^n}{n!} = \frac{1}{n!} \displaystyle\sum\limits_{m=n}^{2n} c_mx^m $
I understand that given $ 0 < x < 1 $ results in
$ 0 < f(x) < \frac{1}{n!} $
but I become confused on . . .
Again $f(0)=0$ and $f^{(m)}(0)=0$ if $m < n$ or $m > 2n.$ But, if $n \leq m \leq 2n $,
$ f^{(m)}(0)=\frac{m!}{n!}c_m $
an integer. Hence $f(x)$ and all its derivatives take integral values at $x=0.$ Since $f(1-x)=f(x),$ the same is true at $x=1.$
All wording kept intact!
The proof that follows actually makes sense when I take for granted the above. I can't however take it for granted as these are, for me, the more important details. So . . .