I have a function of the form

$$ f(k)=\frac{1}{a\_1-a\_{2}k^2e^{-(a\_{3}-a\_{4}k^2)}};\quad k=0...n$$

I approximated it with Taylor series expansion around $k=\frac{n}{2}$, but the results is not very precise (transformation of the function in order to obtain different intervals on the argument, e.g. $[0,1]$ did not work well either). 

Are there other general-purpose polynomial expansions in place that yield sharper bounds?