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Math Help - Monotone sequence

  1. #1
    iva
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    Question Monotone sequence

    Hi there,
    I don't have many example for monotone sequence and have just started a section on them. I think the following is a monotone increasing sequence and would like to check that I'm right:


    (-1)^2 + 2n

    and the reason for that is that:

    Order is preserved ie an < a(n+1)

    Many thanks!
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  2. #2
    Senior Member Tinyboss's Avatar
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    Did you mean (-1)^2? Since that's just 1, I wonder if you meant something else. Maybe (-1)^(2+2n), but that's also just 1. Anyway, a monotone increasing sequence is just one with the property that each term is greater than or equal to the previous term. A strict monotone increasing sequence has every term strictly greater than the previous. And monotone decreasing sequences are defined exactly how you'd think.
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  3. #3
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    (-1)^2 + 2n
    Do you mean (-1)^2 or (-1)^n? If the former, then of course 1 + 2n is a monotone sequence because 1 + 2n < 1 + 2(n + 1). On the other hand, (-1)^n + 2n is monotone, but not strictly monotone. I.e., (-1)^n + 2n <= (-1)^(n + 1) + 2(n + 1) (because changing the sign of 1 may subtract at most 2, but moving from 2n to 2(n + 1) adds 2), but (-1)^2 + 2 * 2 = 5 = (-1)^3 + 2 * 3.
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  4. #4
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    Quote Originally Posted by iva View Post
    the following is a monotone increasing sequence and would like to check that I'm right:
    (-1)^n + 2n
    The first five terms of that sequence are: 1,5,5,9,9,\cdots.
    So it depends upon the definition in your text material.
    There authors (Wall for one) who say that is a monotone non-decreasing sequence. He would have not considered it to be increasing.
    To be increasing Wall says that \left( {\forall n} \right)\left[ {a_n  < a_{n + 1} } \right]
    In fact for your sequence, if n\text{ is even } then a_n=a_{n+1}.
    Does your text address this point at all?
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  5. #5
    iva
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    Monotone

    Thanks for all the feedback guys! I'm sorry there was indeed an error in my equation it is meant to be as Plato put it ie:

    (-1)^n + 2n

    giving the output of 1, 5, 5, 9, 9, 13, 13...

    My textbook ( Fundamentals of Mathematical Analysis by Haggarty) says that a monotone increasing sequence is one in which

    an =< an+1

    So from that I would say that it is an increasing monotone , and it is also a divergent sequence, right?

    Thanks
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  6. #6
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    Quote Originally Posted by iva View Post
    My textbook ( Fundamentals of Mathematical Analysis by Haggarty) says that a monotone increasing sequence is one in which an =< an+1
    So from that I would say that it is an increasing monotone , and it is also a divergent sequence, right?
    Using that definition, you are correct.
    Ii is monotone increasing and not bounded.
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  7. #7
    iva
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    Thank you!
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