The ratio test shows that this series will converge for abs(x/4) <1. The boundary cases x = 4 and x = -4 need to be checked. For x = 4, the series is the harmonic series which does not converge. For x = -4, the series is an alternating sign series which, according to the Leibniz test, does converge (to - ln(2) in fact).
Thus the interval of convergence is - 4 ≤ x < 4. The radius of convergence has to be 4 to match this interval.