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Math Help - monotonic sequence

  1. #1
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    monotonic sequence

    Hi,

    I'm learning about sequences and series in my calc 2 textbook. I have a question about this definition:

    A sequence { a_n} is called increasing if { a_n} < { a_{n+1}} for all n \geq 1. It is called decreasing if { a_n} > { a_{n+1}} for all n \geq 1. It is called monotonic if it is either increasing or decreasing.

    Should that read "if it is *neither* increasing or decreasing." ?

    It looks like a typo, but I'm not sure, so I thought I would ask for clarification. I tried some google searches but couldn't find a clear definition.

    Thanks!
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  2. #2
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    Nope that is correct.
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  3. #3
    MHF Contributor chisigma's Avatar
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    A simple example of sequence that is 'neither increasing nor decreasing' is a_{n}= (-1)^{n}\ \alpha^{n} , \alpha > 0...

    Kind regards

    \chi \sigma
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  4. #4
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    Quote Originally Posted by centenial View Post
    Hi,

    I'm learning about sequences and series in my calc 2 textbook. I have a question about this definition:

    A sequence { a_n} is called increasing if { a_n} < { a_{n+1}} for all n \geq 1. It is called decreasing if { a_n} > { a_{n+1}} for all n \geq 1. It is called monotonic if it is either increasing or decreasing.

    Should that read "if it is *neither* increasing or decreasing." ?

    It looks like a typo, but I'm not sure, so I thought I would ask for clarification. I tried some google searches but couldn't find a clear definition.

    Thanks!
    No, what is says is correct. "Monotonic" means "one direction" which can be either increasing or decreasing. A sequence that is neither, such as 1, 2, 1/2, 3, 4, 5, 1/4, 6, 7. 8, 1/8, ... does not have a special name.
    (A sequence like 1, 1/2, 3/2, 1/4, 7/4, 1/8, 15/8, that alternates increasing is called "alternating".)
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