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Does the series converge or diverge?

  1. Sep 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Determine whether the series [tex]\sum_{n=1}^{\infty} \frac{1}{n+\sqrt{n}}[/tex]
    converges or diverges.
    2. Relevant equations

    3. The attempt at a solution

    This is driving me nuts. I get nowhere with the root test or the ratio test. I can think of some larger series that diverge (no good) and some smaller series that converge (also no good). What test would you recommend?
  2. jcsd
  3. Sep 30, 2009 #2
    The integral test.

    n+√n < n+n = 2n

    [tex]\frac{1}{n + \sqrt{n}} > \frac{1}{2n}[/tex]
  4. Sep 30, 2009 #3
    I'm confused. Doesn't the integral test have us say that if [tex]\int_{1}^{\infty}{\frac{1}{n+\sqrt{n}} dn}[/tex] diverges, then our series diverges?

    Where do you get the 2n from using the integral test?

    Also, the relation [tex]n + \sqrt{n} \leq n+n[\tex] definitely holds, but isn't the series of 1/2n from n=1 to infinity a divergent harmonic series?
  5. Sep 30, 2009 #4
    Ah, well, I suppose in either case, it's a divergent series. This makes me very happy. Thanks.
  6. Sep 30, 2009 #5
    I should have said the the comparison test with the integral test . Sorry about any confusion.

    1/2n diverges, and since it is greater than 1/(n+√n), the latter also diverges.
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