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Show that a Sequence is monotonically decreasing

  1. Dec 9, 2014 #1
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    1. The problem statement, all variables and given/known data

    I was wondering how I would go about showing that (an) is monotone decreasing given that an = 1/√n.

    I believe I have to show an ≥ an+1, but I'm not sure how to go about doing that.
     
  2. jcsd
  3. Dec 9, 2014 #2

    Ray Vickson

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    Well, PF rules require you to make a start on your own. What do you know about the magnitudes of ##\sqrt{n}## and ##\sqrt{n+1}##?
     
  4. Dec 9, 2014 #3
    I know that the magnitude of ##\sqrt{n+1}## is larger than that of ##\sqrt{n}##. Therefore I would assume that the opposite would be true for the magnitude of their reciprocals which would make an≥an+1 as required, however I'm not sure how to write this in a more coherent way.
     
  5. Dec 9, 2014 #4

    pasmith

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    (1) [itex]\sqrt{n + 1} > \sqrt{n}[/itex].
    (2) If [itex]n > 0[/itex] then dividing both sides by [itex]\sqrt{n} > 0[/itex] preserves the inequality. Hence [itex]\frac{\sqrt{n + 1} }{\sqrt{n}} > 1[/itex].
    (3) Dividing both sides by ... > 0 preserves the inequality. Hence ...
     
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