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A factorial limit

Let \cdot! denote the factorial of a real number; that is x!=\Gamma(x+1). Evaluate the limit:

    \[\ell = \lim_{x \rightarrow n} \frac{x!-n!}{x-n}\]

Solution

It holds that

    \begin{align*} \lim_{x\rightarrow n} \frac{x!-n!}{x-n} &= \lim_{x\rightarrow n} \frac{\Gamma(x+1)- \Gamma(n+1)}{x-n} \\ &=\Gamma'(n+1) \\ &=\Gamma(n+1) \psi^{(0)}(n+1) \\ &=n! \left ( \mathcal{H}_n - \gamma \right ) \end{align*}

where \mathcal{H}_n denotes the n-th harmonic number and \gamma the Euler – Mascheroni constant.

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