For a nonnegative real number \(x\), let \[ f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}\] for a positive integer \(n\). Determine \(\lim_{n\to\infty}f_n(x)\).

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For a nonnegative real number \(x\), let \[ f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}\] for a positive integer \(n\). Determine \(\lim_{n\to\infty}f_n(x)\).

For integer \( n \geq 1 \), define

\[

a_n = \sum_{k=0}^{\infty} \frac{k^n}{k!}, \quad b_n = \sum_{k=0}^{\infty} (-1)^k \frac{k^n}{k!}.

\]

Prove that \( a_n b_n \) is an integer.

Let \(a_1\le a_2\le \cdots \le a_n\) be integers. Prove that

\(\displaystyle\prod_{1\le j<i\le n} \frac{a_i-a_j}{i-j}\)

is an integer.