Infinite Sequences and Series · Hard problem

A machine's cumulative wear follows the partial sums $S_N=\displaystyle\sum_{n=1}^N\dfrac1{n!}$ of a series whose total, as $N\to\infty$, is exactly $e-1$ (using $\displaystyle\sum_{n=1}^\infty\dfrac1{n!}=e-1$). After how many terms $N$ does the partial sum $S_N$ first reach at least $99\%$ of that total?