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Homework Help: Product of two eventually increasing sequences which is not eventually increasing

  1. Feb 1, 2012 #1
    1. The problem statement, all variables and given/known data
    Give an example to show that it is not necessarily true that the product of two eventually increasing sequences is eventually increasing.

    2. Relevant equations
    a sequence is eventually increasing if for N[itex]\in[/itex] natural numbers, a[itex]_{n+1}[/itex] [itex]\geq[/itex]a[itex]_{n}[/itex] for all n>N.

    3. The attempt at a solution
    So, I know this is merely proof by counterexample. I find one example to show that the product of two eventually increasing sequences is not necessarily eventually increasing. The only catch is that I have no idea where to start. There are infinitely many eventually increasing sequences I could multiply together. I know the end goal is to show that a[itex]_{n+1}[/itex] - a[itex]_{n}[/itex] is decreasing or eventually decreasing for all n.
    So, ideally, I'd end up with something like -x[itex]^{2}[/itex] after a[itex]_{n+1}[/itex] - a[itex]_{n}[/itex]. I don't want a specific example which will solve this problem. Some guidance as to where to begin would be greatly appreciated, though!
  2. jcsd
  3. Feb 1, 2012 #2


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    Homework Helper

    Try picking an eventually increasing sequence whose values are negative.
  4. Feb 1, 2012 #3
    Sorry that I took an eternity to reply. I was at a study session for linear algebra. Thanks so much, Dick. I think you've helped on every question I've posted here. I really appreciate it. I tripped myself up by only thinking about positive sequences. I took increasing and mistakenly correlated it with positive, too. Immediately after your hint, I thought about -1/n, and then found my solution. This series and sequences course has managed to confuse me more than any previous math class.
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