For example, if you had a series like:

an= for n>=1

what steps would you take to test if it is monotone and bounded?

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- Apr 5th 2010, 11:09 PMMattpdHow do you know if a sequence is monotone and bounded?
For example, if you had a series like:

an= for n>=1

what steps would you take to test if it is monotone and bounded? - Apr 5th 2010, 11:51 PMharish21
Its good to keep in mind the definition of Monotone sequence and bounded sequence.

A sequence is:

(a)**increasing**if and only if for any

(b)**decreasing**if and only if for any

The sequence is**MONOTONE**if either**one**of these properties holds.

So check for the terms in you sequence, and you'll be able to find out if its monotone or not!

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The sequence is**BOUNDED ABOVE**if there exists a number M such that for any . M is called the upper bound.

The sequence is**BOUNDED below**if there exists a number m such that for any . m is called the lower bound.

Can you tackle your question now? - Apr 6th 2010, 04:21 AMHallsofIvy
Is it true that ?

That is, is ?

Since both denominators are positive, that would be the same as

The discriminant of that quadratic is 36- 4(2)(7)< 0 so the quadratic is never 0. In fact, it is easy to see that it is always positive so what is really true is that for all n. Crucially, everystep is "reversible" so we could go from back to . This is a strictly increasing sequence.

To see that it is bounded, note that leads to or 6n^2+ 8< 6n^2+ 9[/tex] and then .

Again, you could start from the obvious fact that and reverse the steps to get . The sequence has 2 as an upper bound.

It should be obvious that this sequence converges to 3/2 and so has 3/2 as "least upper bound". - Apr 6th 2010, 08:49 AMMattpd
I think I have it now, thanks.