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

by oferon
Tags: ann2expsqrtn
 P: 30 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!
 Sci Advisor P: 5,942 Simple method: Let m=√n, so the problem is limit m -> ∞ m4/em. em = 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.
 P: 30 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..