Approximate the value of the series to within an error of at most $10^{-3}$. $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(n+6)(n+2)}$$

According to Equation (2): $$\left| S_N-S \right|\le a_{N+1}$$ what is the smallest value of $N$ that approximates $S$ to within an error of at most $10^{-3}$?
$N =$

$S\approx$