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Quick question on a series

  1. Dec 12, 2011 #1
    1. The problem statement, all variables and given/known data
    [tex]\sum_{n=1}^{\infty}\frac{(-1)^n}{\ln(n+1)}[/tex]


    2. Relevant equations
    absolute convergence test
    nth term test/divergence test


    3. The attempt at a solution
    so the absolute convergence test says that if the absolute value of the series converges then the original series converges absolutely
    so with the series i have, in absolute value is [tex]\sum_{n=1}^{\infty}\frac{1}{\ln(n+1)}[/tex]
    then using the nth term test/divergence test the sequence [itex]a_n=\frac{1}{\ln(n+1)}[/itex] goes to zero as n goes to infinity therefore the series converges, so i have absolute convergence
    but my book says that it only converges conditionally, what am i doing wrong? or is the book wrong?
     
  2. jcsd
  3. Dec 12, 2011 #2

    Dick

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    Science Advisor
    Homework Helper

    The nth term test can only show you that a series diverges. It can't prove it converges. I'd suggest you try a comparison test or an integral test to show 1/ln(n+1) diverges.
     
  4. Dec 12, 2011 #3
    oohhh right, its been awhile for me since ive done problems on series
    cant believe i forgot how the nth term test works, thanks!
     
  5. Dec 12, 2011 #4
    Also, try the alternating series test.
     
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