The value of the series \(4+4x+4x^2+4x^3+\cdots\) is a real number greater than 8. Find all possible values of x, giving your answer in interval notation.
This is a geometric series with a common ratio of x
So we have that....using the sum of an infinite geometric series.....
4 / ( 1 - x) > 8
4 > 8 ( 1 - x)
4 > 8 - 8x
8x > 4
x > 1/2
However x must also be < 1
So the solution is
(1/2 , 1 )
