Is the sequence {a_n} monotone?

[pcvt]

Homework Statement

State whether or not the sequence {a_n} = n+[(-1)^n]/n is monotone or not and justify.

Homework Equations

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.

 

[Mark44]

Homework Statement

State whether or not the sequence {a_n} = n+[(-1)^n]/n is monotone or not and justify.

Homework Equations

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.

What basic axioms are you talking about? Your approach using induction sounds good to me.

 

[pcvt]

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.

 

[Mark44]

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.