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Proving a sequence is monotone.

  1. Nov 4, 2012 #1
    Hi,

    I am unsuccessful at showing that the sequence √n + 1/n is an ascending monotone one, i.e. that a_n+1 > a_n for any n, greater than 2 let's say. I have proven that it is not bounded from above and is bounded from below. Any ideas, suggestions, please?
     
  2. jcsd
  3. Nov 4, 2012 #2

    SammyS

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    This appears to be essentially the same as a previous thread you started:

    https://www.physicsforums.com/showthread.php?t=649117
     
  4. Nov 4, 2012 #3
    Hi,
    Except that no one there was able to help me. Perhaps someone could now.
     
  5. Nov 4, 2012 #4

    Zondrina

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    Consider your sequence as a function.
     
  6. Nov 4, 2012 #5
    I am not allowed to. It has to be proven without any reference to functions. Seemingly merely by showing that a_n+1 > a_n for any n greater than 2. Yet I am unable to show that that inequality holds.
     
  7. Nov 4, 2012 #6

    Zondrina

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    Write out what a_(n+1) actually is.
     
  8. Nov 4, 2012 #7
    sqrt(n+1) + 1/(n+1).
     
  9. Nov 4, 2012 #8

    Zondrina

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    Now what do you do to prove one sequence is larger than the other for all n≥2.
     
  10. Nov 4, 2012 #9
    Induction? I have been unsuccessful at proving it via induction as well.
     
  11. Nov 4, 2012 #10

    Zondrina

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    If [itex]a_{n+1} > a_n \Rightarrow a_{n+1} - a_n > 0[/itex]
     
  12. Nov 4, 2012 #11
  13. Nov 4, 2012 #12

    Zondrina

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    Write out what it means.
     
  14. Nov 4, 2012 #13
    sqrt(n+1) + 1/(n+1) - sqrt(n) - 1/n > 0
     
  15. Nov 4, 2012 #14

    Zondrina

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    You could clean it up a bit, but is that sequence greater than zero for every n?

    If it helps, roughly around n = 1.2, your sequence geometrically will start to be above the line y=0.
     
    Last edited: Nov 4, 2012
  16. Nov 4, 2012 #15
    It surely is greater than zero but I am unable to demonstrate it algebrically, alas.
     
  17. Nov 4, 2012 #16

    Zondrina

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    So, start with n≥2.

    For n=2 does the inequality hold?
     
  18. Nov 4, 2012 #17
    It does.
     
  19. Nov 4, 2012 #18

    Mark44

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