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

  • Thread starter oferon
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In summary, the discussion was about finding the limit of the function n^2e^(-√n) as n approaches infinity. The group determined that the limit is equal to 0, but were struggling to find a formal way to prove it. Various methods were suggested, including using the squeeze theorem and the ratio test, but none seemed to work. Ultimately, the group settled on using the simple method of changing variables and applying Taylor expansion. However, one member suggested using L'Hopital's rule to take derivatives and show that the limit is indeed 0.
  • #1
oferon
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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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  • #2
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.
 
  • #3
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..
 
  • #4
Have learned L'Hopital's rule?
If so, use that. Take 5 derivatives of the numerator and the denominator and get 0/em.
 

1. What is the limit of An=n^2*exp(-sqrt(n)) as n approaches infinity?

The limit of An as n approaches infinity is 0. This can be shown by using L'Hopital's rule or by noticing that the exponential term decreases much faster than the polynomial term increases.

2. How do you find the limit of a sequence?

To find the limit of a sequence, you can either use direct substitution, apply algebraic manipulations to simplify the expression, or use techniques such as L'Hopital's rule or the squeeze theorem.

3. What is the significance of the exponential term in the sequence An=n^2*exp(-sqrt(n))?

The exponential term in the sequence helps to decrease the overall growth of the sequence as n approaches infinity. This results in a limit of 0, rather than the sequence growing infinitely.

4. Can the limit of the sequence An=n^2*exp(-sqrt(n)) be found analytically?

Yes, the limit of the sequence can be found analytically using techniques such as L'Hopital's rule, the squeeze theorem, or algebraic manipulations.

5. Is the sequence An=n^2*exp(-sqrt(n)) convergent or divergent?

The sequence is convergent, as it approaches a finite limit of 0 as n approaches infinity.

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