Probability that n is prime such that n fails the Miller-Rabin test N times
Just one step more, your last expression is

$$\frac{4^N}{4^N +\ln(n)-1}=1-\frac{\ln n -1}{4^N+\ln n -1}\geq 1-\frac{\ln n -1}{4^N}\geq 1-\frac{\ln n}{4^N}.$$