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

  • Context: Graduate 
  • Thread starter Thread starter oferon
  • Start date Start date
Click For Summary

Discussion Overview

The discussion centers around proving the convergence and finding the limit of the sequence defined by \(\lim_{n\to\infty}n^2e^{(-\sqrt{n})}\). Participants explore various methods and approaches to demonstrate this limit, including the squeeze theorem, variable substitution, Taylor expansion, and L'Hôpital's rule.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant asserts that the limit obviously equals 0 but struggles to formally prove it, mentioning difficulties with the squeeze theorem and the ratio test.
  • Another participant suggests a variable substitution \(m=\sqrt{n}\), transforming the limit into \(\lim_{m \to \infty} \frac{m^4}{e^m}\) and argues that the exponential in the denominator grows faster than the polynomial in the numerator.
  • A different participant questions whether using Taylor expansion is the only method to solve the problem, expressing concern that it may not align with expectations from the course staff.
  • Another participant recommends using L'Hôpital's rule, suggesting taking multiple derivatives of the numerator and denominator to evaluate the limit.

Areas of Agreement / Disagreement

Participants present multiple competing views on how to approach the problem, with no consensus on a single method being preferred or correct.

Contextual Notes

Some participants express uncertainty about the appropriateness of certain methods, such as the use of Taylor expansion, and the expectations from course staff regarding the solution approach.

oferon
Messages
29
Reaction score
0
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!
 
Physics news on Phys.org
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.
 
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..
 
Have learned L'Hopital's rule?
If so, use that. Take 5 derivatives of the numerator and the denominator and get 0/em.
 

Similar threads

Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
MHB Prove limit with convergence tests
  • · Replies 2 ·
Replies
2
Views
1K
MHB Is the Convergence of These Sequences Correctly Determined?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad How Does the Ratio Test Determine Convergence in Power Series?
  • · Replies 19 ·
Replies
19
Views
2K
Undergrad What is the limit of (a^n)/n for a>1?
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Limit of ##i^\frac{1}{n}## as ##n \to \infty##
  • · Replies 14 ·
Replies
14
Views
2K
MHB Limit of $x_n$ Sequence: $\pi/2$
  • · Replies 1 ·
Replies
1
Views
2K
MHB No problem, happy to help! (Glad to hear it)
  • · Replies 2 ·
Replies
2
Views
2K