Answer
$n \geq 1178$
Work Step by Step
Since, we have $\int_n^\infty \dfrac{1}{x(\ln x)^{3}}dx \lt 0.01$
or, $\lim\limits_{a \to \infty} (\dfrac{-1}{2(\ln a)^{2}}+\dfrac{1}{2(\ln n)^2})\lt 0.01$
or, $n \gt e^{\sqrt {50}} \approx 1177.405 $
or, $n \geq 1178$
Hence, $n \geq 1178$
