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Homework Help: Real Analysis: proving a sequence converges and finding its limit.

  1. Apr 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose r>1. Prove the sequence [itex]\sqrt[n]{1 + r^{n}}[/itex] converges and find its limit.

    2. Relevant equations

    3. The attempt at a solution

    It's obvious that the sequence converges to r, so I know where I need to end up. My first instinct is to use the squeeze theorem. It's obvious that [itex]\sqrt[n]{r^{n}}[/itex]<[itex]\sqrt[n]{1 + r^{n}}[/itex]. However, I'm having difficulty finding a sequence that's greater than [itex]\sqrt[n]{1 + r^{n}}[/itex] but also converges to r.
  2. jcsd
  3. Apr 24, 2012 #2


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    Try comparing to [itex]\displaystyle \sqrt[n]{2\,r^n}\ .[/itex]
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