I did the second part separately before. Here it is
Assuming that x2-x-1≠1 the finite sum G=Σk=0k=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.