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Homework Help: Determine whether series is convergent or divergent

  1. Aug 4, 2008 #1
    1. The problem statement, all variables and given/known data

    Determine whether or not the series [tex]\sum^{\infty}_{n=1} \frac{1}{\sqrt{n+1}+\sqrt{n}}[/tex] converges.

    3. The attempt at a solution

    Assuming this diverges, I rationalize it to get get [tex]\sum^{\infty}_{n=1} \sqrt{n+1} - \sqrt{n}[/tex]. How would I proceed further?

    Is this even the right approach?
  2. jcsd
  3. Aug 4, 2008 #2
    Well, you can't just assume it diverges. I haven't studied series in quite a while. The rationalizing of the denominators seems like a promising approach, but someone else will have to comment on that.

    I think the following is a valid method. Let f(n) be the expression we are summing. As n-> +inf f(n) -> 1/(2sqrt(n)). But the infinite series with 1/sqrt(n) is a p-series and I think it's divergent.
  4. Aug 4, 2008 #3


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    Science Advisor
    Homework Helper

    snipez90 is right. Don't 'rationalize' it. Just compare it with a divergent p-series.
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