Infinite Sequences and Series · Challenge problem

The tail $\displaystyle\sum_{n=N+1}^{\infty}\frac{1}{n^3}$ of the $p$-series with $p=3$ is bounded above by $\displaystyle\int_N^{\infty}\frac{dx}{x^3}$ (the integral-test remainder estimate). What is the smallest positive integer $N$ for which this bound guarantees that the tail is less than $0.0001$?