# 2014-22 Limit

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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# 2014-09 Product of series

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.

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Let $$a_1\le a_2\le \cdots \le a_n$$ be integers. Prove that
$$\displaystyle\prod_{1\le j<i\le n} \frac{a_i-a_j}{i-j}$$