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Math Help - Monotone sequence - how can this be a monotone sequence if it bounces between numbers

  1. #1
    iva
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    Monotone sequence - how can this be a monotone sequence if it bounces between numbers

    My understanding of showing a sequence is monotone is to show that it either increases or decreases by working out


    (a_n+1) - (a_n)

    and if it's > 0 its increasing, if less than, decreasing, right?

    But now i have this problem that I must say if the sequence is monotone:

    (a_n) = (-1)^n + 2n


    which to me works out that it is bounded by 0 and 4, ie that it bounces between 0 and 4 ( if n is odd then the value is 0 if even then its 4)

    Surely this goes against monotone sequence??? But my answer says that it IS monotone. How can it be monotone when it is neither increasing nor decreasing??
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    A monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing.
    You have a monotonically increasing sequence since a_{n+1}-a_n \geq 0, but it's not strictly monitonically increasing.
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  3. #3
    iva
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    Thanks, I was just replying as I picked up my error (I was confusing the a_n + 1 - a_n with the original sequence. Ok I see it is increasing.

    But what do you mean by "it is not strictly monotonically increasing?"

    Thank you
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    In order for it to be strictly monitonically increasing, you need a_{n+1} - a_n > 0
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