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Absolute convergence help

  1. Oct 23, 2011 #1
    1. The problem statement, all variables and given/known data

    2. Relevant equations
    absolute convergence test

    3. The attempt at a solution
    by book says that the series converges because [tex]\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}[/tex] converges
    but they dont show how the absolute value of the original series converges, and ive tried showing it myself but i keep getting divergence
    i know that as n grows larger and larger the behavior of [tex]\frac{\sqrt{n}+1}{n+1}[/tex] is similar to that of [tex]\frac{\sqrt{n}}{n}[/tex] so i tried using limit comparison and direct comparison with [itex]\frac{1}{n}[/itex] but i keep getting divergence
    i tried the integral test but i kept getting divergence also
    ive been trying this for far too long so any help would be greatly appreciated
  2. jcsd
  3. Oct 23, 2011 #2


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    You are correct that the positive term series diverges.
  4. Oct 23, 2011 #3
    but my book says that the original series converges by the absolute convergence test
    so wouldnt that mean that [tex]\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}[/tex] converges also? or is this an error in the book?
  5. Oct 23, 2011 #4


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    The series is not absolutely convergent. It may be convergent with the alternating signs in which case it would be called "conditionally convergent". (I didn't check that). But the positive term series you are asking about is definitely divergent. You know it is because you correctly checked it.
  6. Oct 23, 2011 #5
    well then ill just check to see if it convergences by the alternating series test
    thanks a lot!
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