Sequences ratio test, intro to real analysis

[Geekster]

Homework Statement

Let X = (x_n) be a sequence of positive real numbers such that lim(x_n+1 / x_n) = L > 1.


Show that X is not a bounded sqeuence and hence is not convergent.

Homework Equations

Definition of convergence states that for every epsilon > 0 there exist some natural number K such that for all n > = K, |x_n - x| < epsilon, then the squence converges to x.

The Attempt at a Solution

This is a proof so there really isn't too much to say here. I have looked at the definition of convergence and I see that I can get something like x_n+1 < x_n(L+e) where e is epsilon, but I do not see any way to produce an upper bound from that. I also know that the previous statement is true for all e, but that still does not seem to get me anywhere. I know I need to use the fact that L > 1, but I don't see how at this point.


Any hints, subtle or not, are welcome.

Thanks,

The Geekster

 

[Dick]

Use epsilon=(L-1)/2. That means for some N, for all n>N. x_n+1>((L+1)/2)*x_n. (L+1)/2>1. So for n>N x_n increases at least as fast as a power series with a ratio > 1.