Series expansion of an integral at infinity

[Irid]

Hello,
I'm fiddling with Wolfram Alpha and I can't find a definition of what do they mean by the "Series expansion of the integral at x -> inf". In particular, I have two divergent integrals and I am wondering whether their ratio is some finite number. Here it is:

\left[\int_0^{\infty} \frac{1}{x^n}e^{1/x}\, dx \right] \left[\int_{-\infty}^{+\infty}e^{u^2} \cos u\, du \right]^{-1}

where n is a parameter. Based on wolfram's suggestion, I think that if n=2, the above expression converges to something meaningful, since both integrals apparently have the series expansion at infinity as exp(x^2).

 

[fresh_42]

I do not see any functions there with $n$ specified. The integrals are definite integrals, i.e. numbers or infinite. Hence the quotient is another number or undefined.

If you manage to make functions out of them, then you can of course write down a Taylor expansion. Radius of convergence is then a different question.