# Sequences ratio test, intro to real analysis

1. Oct 28, 2008

### Geekster

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

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

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

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

3. 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 xn+1 < xn(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

2. Oct 28, 2008

### 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.