\[

\frac{-x}{1+x/\sqrt{n}},

\]

which is decreasing in n both for negative and positive x. ]]>

It is not difficult to verify (via maximization over ), that the integrand is dominated by – which is best possible for (as it attained by the integrand for ).

]]>*[Corrected, thanks – T.]*

I believe I have an answer for my own question above. As $n$ grows the “Gaussian” shape of the integrand t^n exp(-t) narrows around the “center” or peak which is at t=n. Since we are shifting the function to have the origin at t=n. The Taylor approximation makes sense around that location. Yes, the function extendes between -infty and infty, but the gross of the integral is due to the near to the maxima points since in the limit this “Gaussian” shape becomes an spike or some kind of Dirac Delta.

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