I want to approximate the function $ f(x) = x^k e^{-x}$ with some finite series. One approach would be to use the power series expansion for $ e^{-x} $. But in that case, the power series would have to be truncated such that the order of the truncated power series is of order greater than $ x^k $ to ensure that the approximation $ \hat{f}(x) $ does not blow up as $ x $ grows large. But what is the optimum way to choose the order of the truncated power series for $ e^{-x}~~ in ~~ f(x)$?
I couldn't find any useful reference to this problem. Any suggestions?
Are there alternative approaches ?