Consider the following sequence of functions in $L^2[0,\infty)$: $$f_n(x)=\exp\Big(-\frac{x}{n}\Big)x^n,\;\; n=1,2,\cdots$$ Does this sequence span $L^2[0,\infty)$ (that is, is the set of finite linear combinations of these functions dense)?

(My guess is that it doesn't).

Thank you.