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


by oferon
Tags: ann2expsqrtn
oferon
oferon is offline
#1
Feb11-12, 03:28 PM
P: 30
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!
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mathman
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#2
Feb11-12, 04:12 PM
Sci Advisor
P: 5,937
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.
oferon
oferon is offline
#3
Feb11-12, 05:56 PM
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..

mathman
mathman is offline
#4
Feb12-12, 05:53 PM
Sci Advisor
P: 5,937

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


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


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