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How to find the bounds of this siquence

  1. Dec 2, 2008 #1
    (n^2+2)^0.5 - (n)^0.5

    i thought of doing a limit where n->infinity

    but here i get undefined form and even if i whould get some finite limite
    it will only be one bound

    and i cant do limit n->-infinity because its a sequence must be positive??
  2. jcsd
  3. Dec 2, 2008 #2


    Staff: Mentor

    It's not bounded, so you're going to have a tough time finding a bound for it. Of the two terms, the first is dominant and is approximately n. n grows large more quickly than sqrt(n).

    [tex]\lim_{n \rightarrow \infty} \sqrt{n^2 + 2} - \sqrt{n} = \infty[/tex]
  4. Dec 2, 2008 #3
    Factor out a [tex]\sqrt(n^2)[/tex] to see that it increases beyond all positive bounds.

    [tex]\lim_{n \rightarrow \infty} \sqrt(n^2 +2) - \sqrt(n) = \lim_{n \rightarrow \infty}\sqrt(n^2)(\sqrt(1 + 2/n^2) - \sqrt(\frac{1}{n})) [/tex]

    In the second bracket, the first term converges to 1 and the second term converges to 0. So we now have [tex]\lim_{n \rightarrow \infty} \sqrt(n^2)*1 = \infty[/tex]
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