1.05^{n}=2
nlog1.05=log2
n=log2/log1.05=14.20669908
I did the second part separately before. Here it is
Assuming that x^{2}x1≠1 the finite sum G=Σ_{k=0}^{k=N1}(x^{2}x1)k=(1(x^{2}x1)^{N})/(1(x^{2}x1)) .
The only way that this sum converges as N→∞ is for (x^{2}x1)^{N}→0 or that x^{2}x1<1
To force this means that x^{2}x1<1 and x^{2}x1>1
x^{2}x1<1 means that x^{2}x2<0. That is between the roots of this parabola. x^{2}x2=(x+1)(x2)=0 between
x=1 or x=2. So on the interval 1<x<2.; For the second condition x^{2}x1>1, which is x^{2}x>0, that is outside of the interval between the the roots of this parabola. x^{2}x=x(x1)=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.
10/23/2013

Michael F.