How Can You Prove Convergence and Monotonicity in Series and Sequences?

[transgalactic]

http://img209.imageshack.us/my.php?image=14535404zm8.jpg

http://img80.imageshack.us/my.php?image=34587054lt1.jpg

this pictures have a high quality ,you can zoom it.

in this links i show the question and how i tried to solve them.
i was taught this convergence stuff using this type questions.
i solve them like that
---------------
a = (a +6)^0.5
n+1 n

a1=(6)^0.5 n=>1 are given

find if the goes up or down
and if its converges and to what limit??

i know that An+1 and An has the same limit
so i put L (limit) instead of them
and get that there are two possible limits
L=1 L=4

know i want to proove that this series goes up:
i fing A2 using A1=(6)^0.5
A2=(11)^0.5
A2>A1

than using an inequality An+1>An (we assume that it is correct)
i prooved An+2>An+1

and prooved that the function goes up

than the same way i prooved than An<4 (converges to 4).

but in this questions that i added in the link
they don't follow this pattern
i tried to solve them that way
but they are different
how do i solve them??

 

[Dick]

The behavior of the first sequence is closely related to the behavior of the function f(x)=(x^2+2)/3, since a_(n+1)=f(a_n). Eg the sequence will be increasing when a_n is in a region where f(x)-x is positive. Try thinking along those lines. Similarly for 4/(x-5). For the other two remember if you are applying some like the ratio test, the ratio doesn't have to be <1 for ALL n. Just for LARGE n.

 

[HallsofIvy]

In the first problem (the imageshack problem) you are concerned with the convergence of the series (i.e. sum), are you not? You have correctly arrived at the ratio
\frac{a_{n+1}}{a_n}= \frac{n+10}{2n+1}
What is important is the limit of that as n goes to infinity: you can see that easily if you divide both numerator and denominator by n:
\frac{n+10}{2n+1}= \frac{1+\frac{10}{n}}{2+ \frac{1}{n}}
As n goes to infinity, those fractions with n in the denominator go to 0 and you can easily determine whether the entire limit is less than or greater than 1.

 

[transgalactic]

i know that i can make a limit problem from it
but the questions type ask for (as i wrote in the example)

to proove using an inequality if the function is going up or down
and then proove using a second inequality to what limit its going to.

i showed in the example a type of problem
but how i proove in the same way
the other types of questions?