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.