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Divergent Series Proof

  1. Apr 14, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove that the series diverges: [itex]\sum_{i=1}^{\infty}\sqrt{n+1}-\sqrt{n}[/itex]


    3. The attempt at a solution
    I'm trying to use the comparison test, but I don't know what to compare it to. All I have done so far is change the terms to be in fraction form:
    [itex]\sqrt{n+1}[/itex]-[itex]\sqrt{n}[/itex]=[itex]\frac{1}{\sqrt{n+1}+\sqrt{n}}[/itex]

    Any clues on what to do next would be great. Thanks!
     
  2. jcsd
  3. Apr 14, 2014 #2

    Dick

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    Try comparing with [itex]\frac{1}{\sqrt{n+1}+\sqrt{n+1}}[/itex]. Does that converge or diverge? How is it related to your original series?
     
  4. Apr 14, 2014 #3

    Ray Vickson

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    Look at the behavior of ## t_n \equiv \sqrt{n+1} - \sqrt{n}## for large ##n##. It helps to write
    [tex] \sqrt{n+1} = \sqrt{n} \left( 1 + \frac{1}{n} \right)^{1/2} [/tex]
     
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