Convergence of {n/(n^2+1)}: Is it Possible?

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Homework Help Overview

The discussion revolves around the convergence of the sequence {n/(n^2+1)} and whether it approaches a limit as n approaches infinity. Participants are exploring the implications of the sequence's structure in relation to convergence criteria.

Discussion Character

  • Exploratory, Assumption checking, Mixed

Approaches and Questions Raised

  • Some participants attempt to analyze the limit by considering the degrees of the polynomial in the numerator and denominator. Others suggest applying L'Hôpital's rule and multiplying by 1/n to simplify the expression. There is also a distinction made between discussing the sequence versus the series, raising questions about the appropriate methods for each.

Discussion Status

The discussion is active, with participants providing hints and suggesting different approaches to evaluate the convergence of the sequence. There is no explicit consensus on the methods to be used, but several lines of reasoning are being explored.

Contextual Notes

Participants are navigating the distinction between sequences and series, which may affect their approaches to determining convergence. There is an acknowledgment of different mathematical rules applicable to each context.

mmilton
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Homework Statement



Is the sequence {n/(n^2+1)} convergent, and if so, what is it's limit?

Homework Equations


The Attempt at a Solution



I believe it does converge because the higher power is in the denominator, so thus, it's limit is 0.

Any help or hints on if I'm headed in the right direction would be very much appreciated!

Thank you in advance.
 
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You are right, using the rules you've learned about infinity limits will get us ((1/n)/(1+(1/n^2))) and the limit of that as n approaches infinity is 0.
 


mmilton said:

Homework Statement



Is the sequence {n/(n^2+1)} convergent, and if so, what is it's limit?

Homework Equations



The Attempt at a Solution



I believe it does converge because the higher power is in the denominator, so thus, it's limit is 0.

Any help or hints on if I'm headed in the right direction would be very much appreciated!

Thank you in advance.
Multiply the numerator & denominator by 1/n .
 


If you're talking about the SEQUENCE, then it converges. Use a useful little rule known as L'Hôpital.

If you're talking about the SERIES, use the Ratio or Integral tests. It diverges.
 

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