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Math Help - Sequence, Increasing, Not Bounded

  1. #1
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    Sequence, Increasing, Not Bounded

    Hello,

    Can someone explain to me why this is considered not bounded?

    After looking up the definition. I can't seem to understand why this wouldn't be considered bounded.

    If there exists a number m such that for every n we say the sequence is bounded below. The number m is sometimes called a lower bound for the sequence.

    If there exists a number M such that for every n we say the sequence is bounded above. The number M is sometimes called an upper bound for the sequence.
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  2. #2
    Grand Panjandrum
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    Re: Sequence, Increasing, Not Bounded

    Quote Originally Posted by l flipboi l View Post
    Hello,

    Can someone explain to me why this is considered not bounded?

    After looking up the definition. I can't seem to understand why this wouldn't be considered bounded.
    Put:

    a_n=3n+\frac{1}{n}

    Suppose it is bounded above by N >0 which we may suppose an integer, then

    a_n=3n+\frac{1}{n}<N

    Can you see why this is impossible?

    Hint a_n>3n so consider a_N.

    CB
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  3. #3
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    Re: Sequence, Increasing, Not Bounded

    Thanks! I think I get it.
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  4. #4
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    Re: Sequence, Increasing, Not Bounded

    okay, I think I know why. Please correct me if i'm wrong, but the reason why this is not bounded is because there is no limit. It grows as the sequence increases.
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  5. #5
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    Re: Sequence, Increasing, Not Bounded

    it would be better to say it is unbounded, because there is no bound (of course, this also means there is no limit...well, i almost hate to say this, because some people allow "infinte limits" as a valid way of expressing unbounded sequences). as CaptainBlack pointed out, assuming there could BE a bound, leads to a contradiction.

    the idea is, if there is a bound, this bound is some finite number we could (in theory, anyway) actually find. but if no matter what finite number we pick, we can find a term that exceeds that bound, there must not be a bound in the first place (the term number that exceeds a given bound could conceivably be, for some sequences, quite a bit larger than the bound itself. fortunately, in this example, we can use the same "N" term as the bound we try).
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  6. #6
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    Re: Sequence, Increasing, Not Bounded

    Thank you!
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