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Monotonicity of n + (-1)^n/n

  1. Sep 23, 2011 #1
    1. The problem statement, all variables and given/known data
    State whether or not the sequence {a_n} = n+[(-1)^n]/n is monotone or not and justify.


    2. Relevant equations



    3. The attempt at a solution
    It clearly appears to be monotone increasing, so I attempt to prove this. I've tried using induction and tried to prove that {an+1} -{an} >= 0. However, I've got it down to proving n(n-1)>0 for n>1, but I'm not sure how to prove this using just the basic axioms of analysis.
     
  2. jcsd
  3. Sep 23, 2011 #2

    Mark44

    Staff: Mentor

    What basic axioms are you talking about? Your approach using induction sounds good to me.
     
  4. Sep 23, 2011 #3
    Well, would it be possible to redefine a new variable so that one can prove n(n-1)>0 for n>1 using induction? It seems possible to use induction but I'm not sure what to do about the fact that the statement isn't true for n=1.
     
  5. Sep 23, 2011 #4

    Mark44

    Staff: Mentor

    I suppose you could do it by induction, but proving that n(n - 1) > 0 for n > 1 seems too trivial to bother with this technique. One look at the graph of y = x(x - 1) for x > 1 should convince anyone that the inequality is true.

    You could also prove this inequality by noticing that for y = x2 - x has a derivative that is positive for x > 1/2, and that y(0) = y(1) = 0. The graph of this function crosses the x-axis at (1, 0) and increases without bound.

    The expression n(n - 1), where n is an integer, agrees with x(x - 1) for all integer values of x.
     
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