Advanced calculus proof- oscillating sequences

[emira]

Homework Statement

For the sequence defined recursively as follows:

a_1 = 2, and a_(n+1) = 1/ (a_n)^2 for all n from N.

Homework Equations

So, we are supposed to use induction to first fidn if the sequence increases or decreases, and then use induction again to show if it is bounded.

The Attempt at a Solution

If one would take some terms for this sequence, it is easy to see it is oscillating...
a_1 = 2, a_2 = 1/4, a_3 = 16, a_4 = 1/256...so I am stuck trying to prove this is an oscillating sequence using induction. is there any other type of proof to use for this case, because induction seems useless in this case.

If someone has any idea, let me know please.

Thank you in advance for you help!

Emira

 

[sutupidmath]

so you want to show whether this sequence diverges or converges, right?

if this is the case, then from a theorem we know that: let

$$(a_n)$$ be a sequence, then if we can find two subsequences from this one $$(b_n), (c_m)$$ such that they don't converge at the same place, then we can conclude that the original sequence a_n diverges. So, all you need to do is let

$$(b_n), for, n=2k, k \in Z^+$$ and $$(c_m), m=2k+1, k\in Z^+$$ and prove that these two subsequences do not converge at the same place.

 

[Dick]

Induction isn't useless, you just have to use it wisely. a_(n+2)=a_n^4, right? The odd terms are increasing (since a1>1) and the even terms are decreasing since (a2<1). That's not to hard to show by induction, right?

 

[HallsofIvy]

Looking at even n, a₂= 1/4, a₄= 1/256, etc. which looks like a decreasing sequence.

Looking at odd n, a₁= 2, a₃= 8, etc. which looks like an increasing sequence.

You know a formula to get a_n+1 from a_n. Can you extend it to get a formula for a_{n+2[/sup] from an? Then use induction on the even and odd subsequences.}