Proving Divergence of (x_n): A Homework Challenge

[ILikePizza]

Homework Statement

(x_n) is a sequence and x_1 > 2. From then on, x_{n+1} = x_n + 1/x_n

Prove that (x_n) is divergent.

Homework Equations

n/a

The Attempt at a Solution

I first tried assuming that a limit existed, but I didn't get a contradiction. (I had x = 2 + 1/x, x = (2 \pm \sqrt{5})/2, which could make sense.)

thankS!

 

[statdad]

You can immediately rule out the possibility

$$\frac{2 - \sqrt 5} 2$$

since that is a negative number and all terms in the sequence are positive.

to show it doesn't converge show

1) The sequence is monotone increasing (easy, you need to show $$a_{n+1} - a_n > 0$$ for every $$n$$
2) Show that for every $$n$$ it is true that $$x_n > n$$ - this will show that the sequence is not bounded above

these two points will show the sequence does not converge

 

[ILikePizza]

Thanks for the reply.

I am aware that this will work, but I still have no idea where to start.

It is easy to show that this sequence is monotone increasing, but what about the second part? That's where I'm stuck.

Thanks

 

[statdad]

Think along these lines

You are told that $$x_1 > 2$$

Consider the function

$$f(x) = x + 1/x$$

Calculate $$f'(x)$$
* What sign does this have for $$x > 2$$?
* Is $$f$$ continuous for $$x > 2$$? Is it bounded?
* What does the sign of $$f'(x)$$ say about the behavior of $$f$$?
* Note that $$x_{n+1} = f(x_n)$$

 

[Dick]

Ok, you know the sequence is monotone increasing and always bigger than 2. If the sequence were bounded above then you know it would have a limit. If it had a limit, the limit would have to satisfy L=L+1/L (not L=2+1/L). So?

 

[ILikePizza]

L+1/L = x_n = L
x _n+1 > L,

a contradiction? Is that what you were fishing for?

 

[Dick]

If x_n has a limit L, then the limit of x_(n+1) is L and the limit of x_n+1/x_n is L+1/L. Are there any solutions at all to L=L+1/L?? I don't think there are. So yes, that's a contradiction. What do you conclude from the presence of a contradiction. Which assumption must be wrong? That's what I'm fishing for. Bite.