Lim of An=n^2*exp(-sqrt(n))

oferon

Hi all, my problem regards this limit:

[tex]\lim_{n\to\infty}n^2e^{(-\sqrt{n})}[/tex]

Obviously equals 0, but I can't find how to show it.
Tried the squeeze theorem (coudn't find any propriate upper bound)
Ratio test won't seem to work..
I do realize the reason for that is that the set approaches 0 starting at heigher n's..

Anyway.. how can I prove convergence and find the limit in a formal way? thanks!

mathman

Simple method: Let m=√n, so the problem is limit m -> ∞ m⁴/e^m.

e^m = 1 + m + m²/2! + m³/3! + m⁴/4! + m⁵/5! + .... It is obvious from the 5th term on the denominator of the fraction swamps the numerator.

oferon

I've tried changing variables like you did and got m4/em, which does seem nicer..
But is using taylor expansion the only way to solve here?
I'm pretty sure that's not what the course staff expected us to do..

mathman

Have learned L'Hopital's rule?
If so, use that. Take 5 derivatives of the numerator and the denominator and get 0/e^m.