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Math Help - Example of sequence with given property

  1. #1
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    Example of sequence with given property

    What examples would be true for the following?

    1. The sequence is convergent but not monotone
    2. The sequence is strictly decreasing but not convergent
    3. The sequence is neither bounded nor monotone
    4. The sequence is bounded below, not bounded above, and contains an infinite number of negative terms
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Come on, these can't be that hard.

    Quote Originally Posted by summerset353 View Post
    What examples would be true for the following?

    1. The sequence is convergent but not monotone
    a_n=\frac{(-1)^n}{n}

    2. The sequence is strictly decreasing but not convergent
    Think about, a_n=-\left(1+\frac{1}{n}\right)^n for example in \mathbb{Q}

    3. The sequence is neither bounded nor monotone
    n(-1)^n


    4. The sequence is bounded below, not bounded above, and contains an infinite number of negative terms
    how about a_n=\begin{cases} n &\mbox{if}\quad n\text{ is even}\\ \frac{-1}{n} & \mbox{if} \quad n\text{ is odd}\end{cases}. Clearly a_n\geqslant1 but is not bounded above and contains infinitely many negative terms.
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  3. #3
    Super Member redsoxfan325's Avatar
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    Even easier for 2: x_n=-n.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by redsoxfan325 View Post
    Even easier for 2: x_n=-n.
    I know. I just was trying to make realize that saying a sequence is convergent without speaking of the ambient space is just...well wrong.
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  5. #5
    Super Member redsoxfan325's Avatar
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    Lol...I agree.
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