Answer
$n \geq 1178$
Work Step by Step
Consider $\int_n^\infty \dfrac{1}{(x)(\ln x)^{3}}dx \lt 0.01$
and $ \lim\limits_{k \to \infty} [\dfrac{(-1)}{(2)(\ln (k))^{2}}+\dfrac{1}{(2)(\ln n)^2}\lt 0.01$
Thus, $n \gt e^{(\sqrt {50})} \approx 1177.405 \implies n \geq 1178$
Hence, $n \geq 1178$
