.
An infinite geometric progression
a, ar, ar^2, ar^3, . . .
has a sum if and only if its common ratio "r" has the modulus (= the absolute value) |r| less than 1: |r| < 1.
In your case r = 4x, so the condition that this geometric series does converge is
|4x| < 1, or, equivalently, |x| < 
, or -0.25 < r < 0.25.
