# How to show sequnce is divergent (math. induction)

1. Oct 3, 2012

### sitia

1. The problem statement, all variables and given/known data

For the sequnce an defined recursively, I have to determine the limiting value, provided that it exists.
a1=2 and a(n+1) = 1/(an)^2 for all n

2. Relevant equations

3. The attempt at a solution

Ok, so I 've done problems like this but the a(n+1) was biggger than an and so the sequence was increasing and I used mathematical induction and boundness to show it converged. Then I founf the limit.
Now, though I'm having trouble with this sequence becasue it's a decreasing sequence and I know it diverges. How do I go about proving that? What method do I use? At what point is it realized that it diverges?
I'm just really lost :/

Thank you!

2. Oct 3, 2012

### SammyS

Staff Emeritus
Have you written out the first several terms of the sequence?

3. Oct 3, 2012

### sitia

I have....1/4, 1/16...etc

4. Oct 3, 2012

### Staff: Mentor

a1 = 2
a2 = 1/(22) = 1/4
a3 = ?
Hint: It's not 1/16.

5. Oct 3, 2012

### sitia

Isn't it 1/(2^2)^2

6. Oct 3, 2012

### Staff: Mentor

Not according to what you wrote in post 1.

7. Oct 3, 2012

### SammyS

Staff Emeritus
"It", as in a3 is

$\displaystyle a_3=\frac{1}{{a_2}^2}$
$\displaystyle =\frac{1}{(1/4)^2}$

$\displaystyle =\frac{1}{1/16}$

$= \underline{\ \ ?\ \ }$

8. Oct 3, 2012

### Staff: Mentor

You shouldn't say "it" unless it is crystal clear what the antecedent is.

9. Oct 3, 2012

### sitia

Oh wow, stupid mistake. Thanks!
So, do I need to show anything to show divergence or just state that it's going to zero and infinity..oscillating, so it's divergent?

10. Oct 3, 2012

### happysauce

Well there certainly are two subsequences that converge to 0 and diverge to infinity.