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

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

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# 1 Answer

Assuming that x

^{2}-x-1≠1 the finite sum G=Σ_{k=0}^{k=N}^{-1}(x^{2}-x-1)^{k}=(1-(x^{2}-x-1)^{N})/(1-(x^{2}-x-1)) .The only way that this sum converges as N→∞ is for (x

^{2}-x-1)^{N}→0 or that |x^{2}-x-1|<1To force this means that x

^{2}-x-1<1 and x^{2}-x-1>-1x

^{2}-x-1<1 means that x^{2}-x-2<0. That is between the roots of this parabola. x^{2}-x-2=(x+1)(x-2)=0 betweenx=-1 or x=2. So on the interval -1<x<2.; For the second condition x

^{2}-x-1>-1, which is x^{2}-x>0, that is outside of the interval between the the roots of this parabola. x^{2}-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.