Problem 11

[Analysis]

According to the properties of the arithmetic sequence and the formula for the sum of the first $n$ terms, we have $(S_{1\,008} - S_{1\,007}) > 0$, which implies $a_{1\,008} > 0$. From $S_{1\,006} > S_{1\,008}$, we get $(S_{1\,008} - S_{1\,006}) 0$ and $S_{2\,014} = \frac{2\,014(a_1 + a_{2\,014})}{2} = \frac{2\,014(a_{1\,007} + a_{1\,008})}{2} 0$, which implies $a_{1\,008} > 0$.

From $S_{1\,006} > S_{1\,008}$, we get $(S_{1\,008} - S_{1\,006}) 0$ and $S_{2\,014} = \frac{2\,014(a_1 + a_{2\,014})}{2} = \frac{2\,014(a_{1\,007} + a_{1\,008})}{2} < 0$.

Therefore, the positive integer value of $n$ that satisfies $\boxed{S_n S_{n-1} < 0}$ is $n = \boxed{2\,015}$.