I want to find a specific range of $\alpha$ formula as follows.
$$\alpha =\frac{1+\frac{{{x}^{2}}}{3!}+\frac{{{x}^{4}}}{5!}++\cdots+{\frac {x^{2n}}{(2n+1)!}}}{1+\frac{{{x}^{2}}}{2!}+\frac{{{x}^{4}}}{4!}+\cdots +{\frac {x^{2n}}{(2n)!}}}$$
We know that 0 < $\alpha$ < 1. But, I want to narrow this range or make it more specific value as much as possible? If it is possible, could you please suggest me some ways to do it without depending on the variable $n$ ($n\to \infty $)?
