For example, if you had a series like:
an= $\displaystyle (3n^2+4)/(2n^2+3)$ for n>=1
what steps would you take to test if it is monotone and bounded?
Its good to keep in mind the definition of Monotone sequence and bounded sequence.
A sequence $\displaystyle {a_n}$ is:
(a) increasing if and only if $\displaystyle {a_n}<{a_{n+1}}$ for any $\displaystyle n \geq 1$
(b) decreasing if and only if $\displaystyle {a_n}>{a_{n+1}}$ for any $\displaystyle n \geq 1$
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 $\displaystyle {a_n} \leq M$ for any $\displaystyle n \geq 1$. M is called the upper bound.
The sequence is BOUNDED below if there exists a number m such that $\displaystyle {a_n} \geq m$ for any $\displaystyle n \geq 1$. m is called the lower bound.
Can you tackle your question now?
Is it true that $\displaystyle a_{n+1}\le a_n$?
That is, is $\displaystyle \frac{3(n+1)^2+ 4}{2(n+1)^2+ 3}\le \frac{3n^2+ 4}{2n^2+ 3}$?
Since both denominators are positive, that would be the same as $\displaystyle (3(n+1)^2+ 4)(2n^2+ 3)\le (2(n+1)^2+ 3)(3n^2+ 4)$
$\displaystyle (3n^2+ 6n+ 7)(2n^2+ 3)\le (2n^2+ 4n+ 5)(3n^2+ 3)$
$\displaystyle 6n^4+ 12n^3+ 23n^2+ 18n+ 21\le 6n^3+ 21n^2+ 12n+ 15$
$\displaystyle 2n^2+ 6n+ 7\le 0$
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 $\displaystyle 2n^2+ 6n+ 7> 0$ for all n. Crucially, everystep is "reversible" so we could go from $\displaystyle 2n^2+ 6n+ 7> 0$ back to $\displaystyle a_{n+1}> a_n$. This is a strictly increasing sequence.
To see that it is bounded, note that $\displaystyle \frac{3n^2+ 4}{2n^2+ 3}< 3/2$ leads to $\displaystyle 2(3n^2+ 4)< 3(2n^2+ 3)$ or 6n^2+ 8< 6n^2+ 9[/tex] and then $\displaystyle 8< 9$.
Again, you could start from the obvious fact that $\displaystyle 8< 9$ and reverse the steps to get $\displaystyle \frac{3n^2+ 4}{2n^2+ 3}< 3/2$. 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".