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Math Help - Sequence Convergence

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    Sequence Convergence

    I have two sequences:

    {0,1,0,0,1,0,0,0,1...}

    {\frac{1}{1},\frac{1}{3},\frac{1}{2},\frac{1}{4},\  frac{1}{3},\frac{1}{5},\frac{1}{4},\frac{1}{6},...  }

    By the Monotonic Sequence Theorem, I think that these are both convergent. The first sequence is bounded between 1 and 0 and is monotonic. Therefore it must converge right? But where? At 0?

    The same seems to be true with the second sequence. The second sequence is also bounded between 1 and 0. It doesn't fit the definition of decreasing or increasing since it jumps up and down, so it must be monotonic. Yet it's gradually approaching a value of zero.
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    Quote Originally Posted by adkinsjr View Post
    I have two sequences:

    {0,1,0,0,1,0,0,0,1...}

    {\frac{1}{1},\frac{1}{3},\frac{1}{2},\frac{1}{4},\  frac{1}{3},\frac{1}{5},\frac{1}{4},\frac{1}{6},...  }

    By the Monotonic Sequence Theorem, I think that these are both convergent. The first sequence is bounded between 1 and 0 and is monotonic. Therefore it must converge right? But where? At 0?

    The same seems to be true with the second sequence. The second sequence is also bounded between 1 and 0. It doesn't fit the definition of decreasing or increasing since it jumps up and down, so it must be monotonic. Yet it's gradually approaching a value of zero.
    What makes you think that the first is monotonic?
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    Quote Originally Posted by Drexel28 View Post
    What makes you think that the first is monotonic?
    Ok, I actually misread the definition of monotonic. A sequence is monotonic if it is either increasing or decreasing. I accidently read that as Neither

    So it's not monotonic. It doesn't have a limit, so it is divergent correct?

    But for the second sequence, it doesn't seem to fit the definition of monotonic either. Even though it is decreasing in general, it jumps up and down. So I can't say that a_n>a_{n+1},\forall n.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by adkinsjr View Post
    Ok, I actually misread the definition of monotonic. A sequence is monotonic if it is either increasing or decreasing. I accidently read that as Neither

    So it's not monotonic. It doesn't have a limit, so it is divergent correct?

    But for the second sequence, it doesn't seem to fit the definition of monotonic either. Even though it is decreasing in general, it jumps up and down. So I can't say that a_n>a_{n+1},\forall n.
    You tell me. Why do you think it converges/diverges?
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    Quote Originally Posted by Drexel28 View Post
    You tell me. Why do you think it converges/diverges?
    I'm undecided now. The reason I originally though that it was convergent was by the Monotonic Sequence Theorem. However, I misread the definition of monotonic, so I now know that it is not monotonic. So the theorem doesn't apply.
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