# Convergence questions

1. Nov 9, 2007

### 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).

i tried to solve them that way
but they are different
how do i solve them??

2. Nov 9, 2007

### 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.

Last edited: Nov 9, 2007
3. Nov 9, 2007

### HallsofIvy

Staff Emeritus
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.

4. Nov 9, 2007

### 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?