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Math Help - Question about monotone sequences.

  1. #1
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    Question about monotone sequences.

    The question is:

    Let ( a_n) and ( b_n) be monotone sequences. Prove or give a counterexample.

    a. The sequence ( c_n) given by ( c_n)= k* a_n is monotone for any real number k.

    I don't think the change of sign on a constant will do more than turn a monotone increasing sequence to a monotone decreasing sequence so this statement is probably true.


    b. The sequence ( c_n) given by ( c_n)= a_n/[tex]b_n[tex] is monotone.

    I'm also pretty sure that having opposite signs for the sequences being divided isn't going to change much either. So I suspect I need to show the different cases on this proof is that the general idea?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Question about monotone sequences.

    The statement a. is true. However, b. is false (even supposing a_n/b_n well defined i.e. b_n\neq 0 for all n ) . Choose for example a_1=1,a_2=2 , a_n=3 for all n\geq 3 and b_1=1 , b_n=3 for all n\geq 2 .
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