HMMT 2016 U19

Let

$$A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}}$$

Find the largest integer less than or equal to $\frac{1}{A}$. The following decimal approximation might be useful: $0.6931<\ln (2)<0.6932$, where $\ln$ denotes the natural logarithm function.