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What am I doing wrong? (Monotonicity of sequence)

  1. Aug 12, 2008 #1
    1. The problem statement, all variables and given/known data
    Determine the monotonicity of the sequence with An as indicated

    An = (-1)^(2n+1) * n^0.5

    3. The attempt at a solution

    well without even doing any tests, I can see that 2n+1 is always odd, so the -1 will make every term negative
    so it simplifies to
    An = - root(n)
    which is obviously a decreasing sequence.....

    Now here is where my problem somes in
    If i do a ratio test of An+1 / An I get

    An+1 / An = [ (-1)^(2(n+1)+1) * root (n+1) ] / [ (-1)^(2n+1) * root (n) ]
    = [ (-1)^(2n + 3) * root (n+1) ] / [ (-1)^(2n+1) * root (n) ]

    when i divide (-1)^(2n + 3) by [ (-1)^(2n+1) i get (-1) ^ 2 which is 1
    so An+1 / An reduces to = root (n+1) / root (n) which is always greater than 1.... now if its greater than one, doesn't it mean that each term is greater than th last, and the seqence is increasing? but from simplifying the expression for the term An I can clearly see that it hs to be decreasing...so where am I screwing up? Am i not supposed to apply ratio tests to sequences with each term being negative? or does the ratio test work the opposite way with sequences that are always negative (ie i need it to be less than one for it to be increasing and more than 1 for it to be decreasing?) This is probably a really dumb question.
  2. jcsd
  3. Aug 12, 2008 #2


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    Homework Helper

    Write out the first three or four terms of [tex] A_n [/tex] explicitly to see what the terms
    look like. That will show you whether the sequence is monotone increasing, decreasing, or neither.
    Once you believe you know which it is, you need to prove it. Examine

    A_{n+1} - A_n

    If this is positive you have one answer, if it is negative you have another.

    Hint: what do you know about [tex] (-1)^{2n+1}[/tex] ?
  4. Aug 12, 2008 #3
    Well I understand the reasoning and i know what it tells me about the -1 (i mention it in the post) so I can see that the sequence is decreasing just from lookin at the simplified form . My question is more why the ratio test fails, whether the ratio test can only be applied to sequens with all positive terms, or whether i can be applied to negative sequences but in reverse (ie An+1/An < 1 means increasing)
  5. Aug 12, 2008 #4


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    Homework Helper

    Sorry - I simply missed part of your question.
    You will find the Ratio Test stated in several cosmetically different, yet equivalent ways, but all of them deal with the problem of investigating convergence of a series, not a sequence. Further, when the ratio test is used, it is specified that for the series

    \sum_{n=1}^\infty a_n

    the terms [tex] a_n [/tex] are all positive , or the test is performed with the absolute values of the terms, as

    \lim_{n \to \infty} \left\vert \frac{a_{n+1}}{a_n} \right\vert

    The ratio test doesn't apply to your question because there is no question of convergence, as you do not have a series: you wanted to know something about the monotonicity of the sequence.

    Since the powers of [tex] -1 [/tex] cancel out in the ratio of terms, your calculation showed that the magnitude of [tex] A_n [/tex] increases as [tex] n [/tex] increases. This, coupled with the fact that all terms are negative, gives the answer to the monotonicity question.

    Sorry for the wordiness, and for missing part of your original question.
  6. Aug 12, 2008 #5
    Thanks so much :) That makes sense
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