Sequences - Assumption that I need to confirm about n approaching infinity

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    Infinity Sequences
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Homework Help Overview

The discussion revolves around finding the limit of the sequence defined by an = 2n/(n² + 1)^(1/2) as n approaches infinity. Participants are exploring the implications of simplifying the expression by disregarding certain terms in the denominator.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster questions whether it is acceptable to disregard the +1 in the denominator when n approaches infinity, suggesting that this simplification leads to a limit of 2. Other participants confirm this approach and discuss the reasoning behind it.

Discussion Status

The discussion is active, with participants providing feedback on the original poster's reasoning. Some guidance has been offered regarding the simplification process, and there is a shared understanding of the approach being valid, though no consensus on the overall methodology has been reached.

Contextual Notes

Participants are considering the behavior of the sequence as n becomes very large, specifically questioning the relevance of the constant term in the denominator. There is an emphasis on understanding the limit properties without reaching a definitive conclusion.

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Homework Statement


Find the limit:
an= 2n/(n2+1)1/2

Homework Equations


n/a

The Attempt at a Solution


Because n is approaching infinity, is it OK to disregard the +1 in the denominator and just consider the denominator to be n? This would then divide out the n in the numerator leaving 2 which is the correct answer. I think this is acceptable, but I wanted to run it by you all to confirm. Thank you in advance for your help.
 
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Yes you can consider the limit like that, the '1' becomes negligible as n gets bigger and bigger.
 
great, that was what I thought but I wanted to make sure that what I got and the correct answer weren't just a great coincidence. Thanks for the fast reply! :)
 
factor out an n^2 in the denominator in the square root to get: sqrt((n^2)(1+(1/n^2))).

then the problem should be something like: (2n)/[(n)(sqrt(1 + (1/n^2)))].

Apply the properties of a limit taken to infinity and you should get 2 as your limit.

So in essence, you could ignore the 2, but to show why you can ignore it, you can do what I just showed you.
 

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