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Determine if a sequence {an} is monotonic, bounded, convergent

  1. Jan 12, 2008 #1
    Determine whether the sequence {an} defined below is
    (a) monotonic
    (b) bounded
    (c) convergent and if so determine the limit.

    (1) {an}=(sqrt(n))/1000
    a) it is monotonic as the sequence increase as n increases.
    b) it's not bounded (but i'm not sure why)
    c) divergent since limit doesnt exit (tends to infinity)
    is this correct?
    (2) {an}=(-2n^2)/((4n^2)-1)
    i dont know how to answer parts (a) (b) and (c) to this one can someone help me please.
    thank you
  2. jcsd
  3. Jan 14, 2008 #2

    Gib Z

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    Homework Helper

    1) a) Correct.

    b) A sequence is bounded if |a_n| is less than some real number M, for all values of n. Does this help? Is there such a number M in this case?

    c) Correct. Also, an unbounded monotonic increasing sequence necessarily diverges to positive infinity. It may be a good exercise to prove why.

    2) a) Trick is to subtract a half and then add it again, like this;

    [tex]a_n = - \left( \frac{2n^2}{4n^2 -1} \right) = - \left( \frac{2n^2 - \frac{1}{2} }{4n^2 -1} + \frac{1}{2(4n^2-1)}\right) = - \left ( \frac{1}{2} + \frac{1}{2(4n^2-1)}\right)[/tex].

    Now use partial fractions, it should be much easier then =]

    PS - I noticed its been about one and a half days since you've posted this, and yet received no help. This is probably because you placed this in the wrong forum - Put it in the Homework Help forums (Calculus and Beyond) next time and you should get help much faster.
  4. Jan 14, 2008 #3
    A common trick with powers in a fraction is to divide up and down by the highest power, like this:
    [tex]a_n = -\frac{\frac{2 n^2}{n^2}}{\frac{4 n^2 - 1}{n^2}} = -\frac 2 {4 - \frac 1 {n^2}}[/tex]​
    Now it's easy to see what the limit is.
  5. Jan 14, 2008 #4

    Gib Z

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    Homework Helper

    Shoot me that is a much easier way :(
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