If the common ratio of a GP with a sum to infinity is x^2-x-1, within what limits must lie?

Question

I feel we have two unknown values here, either the first term or sum to infinity should have been given. I think so! Help

Answer 1

Assuming that x²-x-1≠1 the finite sum G=Σ_k=0^k=N-1(x²-x-1)^k=(1-(x²-x-1)^N)/(1-(x²-x-1)) .

The only way that this sum converges as N→∞ is for (x²-x-1)^N→0 or that |x²-x-1|<1

To force this means that x²-x-1<1 and x²-x-1>-1

x²-x-1<1 means that x²-x-2<0.  That is between the roots of this parabola. x²-x-2=(x+1)(x-2)=0 between

x=-1 or x=2.  So on the interval -1<x<2.;  For the second condition x²-x-1>-1, which is x²-x>0, that is outside of the interval between the the roots of this parabola.  x²-x=x(x-1)=0 when x=0 or x=1. So on the intervals (-∞,0) or (1,∞)

The intersection of these intervals is (-1<x<0)∪(1<x<2).  For x in this set the infinite sum converges.