Proving a monotonic sequence is unbounded

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Discussion Overview

The discussion revolves around proving that a specific sequence defined by the recurrence relation \(x_{n+1} = x_n + \frac{1}{x_n^2}\) is unbounded. Participants explore various approaches to this proof, including the implications of boundedness and convergence.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes proving by contradiction, assuming the sequence is bounded and trying to derive a contradiction regarding its monotonicity.
  • Another participant states that if the sequence were bounded, it would converge.
  • A further contribution suggests that if the sequence converges to some limit \(x\), then it must satisfy the equation \(x = x + \frac{1}{x}\), prompting questions about the values of \(x\) that could satisfy this equation.
  • Another participant expresses confusion about the reasoning behind the limit argument and its implications for convergence.
  • A later reply suggests taking the limit of both sides of the recurrence relation to analyze convergence, indicating that if the sequence converges, both \(x_n\) and \(x_{n+1}\) would approach the same limit \(x\).

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are differing interpretations of the implications of boundedness and convergence, and some express confusion about the reasoning presented.

Contextual Notes

There are unresolved assumptions regarding the behavior of the sequence under the conditions of boundedness and convergence, as well as the implications of the derived limit equation.

alligatorman
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I'm trying to prove that the sequence
[tex]x_1,x_2,_\cdots[/tex]
of real numbers, where
[tex]x_1=1[/tex] and [tex]x_{n+1}=x_n+\frac{1}{x_n^2}[/tex] for each [tex]n=1,2, \cdots[/tex]

is unbounded.

(sorry for the ugly latex! i don't know if there's a way to format that better)

I'm thinking of proving by contradiction, assuming it is bounded and then somehow getting it to imply that the sequence is not increasing, but I'm not sure how to go about it.

Any hints?
 
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If it were bounded, it would converge.
 
And if it were to converge to, say, x, that limit would satisfy
[tex]x= x+ \frac{1}{x}[/tex]
What values of x satisfy that?
 
HallsofIvy said:
And if it were to converge to, say, x, that limit would satisfy
[tex]x= x+ \frac{1}{x}[/tex]
What values of x satisfy that?

:confused: I don't understand why this is true.
 
Start with [itex]x_{n+1}= x_n+ 1/x_n[/itex] and take the limit, as n goes to infinity ,of both sides. If the sequence [itex]{x_n}[/itex] converges to some number, x, then each "[itex]x_n[/itex]" or "[itex]x{n+1}[/itex]" term will go to x.
 

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