Does the series converge or diverge?

  • Thread starter quasar_4
  • Start date
  • #1
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Homework Statement



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

Homework Equations





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?
 

Answers and Replies

  • #2
867
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The integral test.

n+√n < n+n = 2n

[tex]\frac{1}{n + \sqrt{n}} > \frac{1}{2n}[/tex]
 
  • #3
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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?
 
  • #4
290
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Ah, well, I suppose in either case, it's a divergent series. This makes me very happy. Thanks.
 
  • #5
867
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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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