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Alternating Series Help

  1. Nov 21, 2005 #1
    I apologize right now for the fact that I have no idea how to use LaTeX
    I can't figure out if the following alternating series is convergent or not:
    Sum(((-1)^(n-1)) * ((2n+1)/(n+2))) from 1 to infinity
    the root test is not applicable, A(n+1)>An, and the ratio test gives me Limit=1, so I have no comclusive evidence either way. Even Maple 10 couldnt give me an answer.
    I have a test tomorrow. HELP!!!!!!!
  2. jcsd
  3. Nov 21, 2005 #2


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    Since it is alternating, you can combine terms pairwise. You will then get a monotonic series with terms ~2/n for large n. This is, as you should know, divergent.
  4. Nov 21, 2005 #3


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    [tex]\sum_1^\infty (-1)^{n-1} \frac{2n+1}{n+2}[/tex]
    Is an alternating series. Therefore it converges if, and only if, the limit of the individual terms goes to zero.
    [tex]\lim_{n \rightarrow \infty} (-1)^{n-1} \frac{2n+1}{n+2}[/tex]
    does not exist. (There are limit points at +2 and -2.) Since the sequence of terms does not converge, the series cannot converge.
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