How can I prove convergence and find the limit of this sequence?

[oferon]

Hi all, my problem regards this limit:

\lim_{n\to\infty}n^2e^{(-\sqrt{n})}

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.