Question 114
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: $0.6931 \lt \ln(2) \lt 0.6932$, where $\ln$ denotes the natural logarithm function.
$$
A = \lim_{n \rightarrow \infty} \sum_{i=0}^{2016} (-1)^i \cdot \frac{\binom{n}{i} \binom{n}{i+2}}{\binom{n}{i+1}^2}
$$
Find the largest integer less than or equal to $\frac{1}{A}$.
The following decimal approximation might be useful: $0.6931 \lt \ln(2) \lt 0.6932$, where $\ln$ denotes the natural logarithm function.