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Why does this infinite series diverge?

  1. Nov 5, 2011 #1
    1. The problem statement, all variables and given/known data
    Why does this series diverge?


    2. Relevant equations
    [itex]\sum_{n}^{\infty }\frac{-1^{(2n+2)}}{n+1}[/itex]


    3. The attempt at a solution

    I must be missing a rule with the -1 sign.

    My logic is that for all n, the numerator = 1 since anything to the power of 2 is even and adding the 2 doesn't change the sign either. So if the numerator is always 1 and the denominator grows without bound so by the alternating series test since, L = 0 and the series is decreasing, the series should converge... My math book and maple says different though :(
     
    Last edited: Nov 5, 2011
  2. jcsd
  3. Nov 5, 2011 #2

    SammyS

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    As you pointed out, it's not an alternating series.

    For clarity, put parentheses around the -1. _ _ _ _ (-1)

    [itex]\displaystyle \sum_{n}^{\infty }\frac{(-1)^{(2n+2)}}{n+1}[/itex]
     
  4. Nov 5, 2011 #3
    S=Ʃ(-1)2k+2/(k+1)
    Try a simpler case of the above. Try the other tests. Maybe the integral test esp. something looking like a log fn differentiated.
     
    Last edited: Nov 5, 2011
  5. Nov 5, 2011 #4
    I think I have it... If the numerator is positive for all n, I can just use the direct comparison test with B sub N = 1/n which is a divergent p-series and since B sub N is less than A sub N, A sub N also diverges... right?
     
  6. Nov 5, 2011 #5

    SammyS

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    That sounds good to me !
     
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