Limit to Infinity of x minus Gamma of Reciprocal of x
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Theorem
- $\ds \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\dfrac 1 x} } = \gamma$
where:
- $\Gamma$ denotes the $\Gamma$ (Gamma) function
- $\gamma$ denotes the Euler-Mascheroni constant.
Proof
\(\ds \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\frac 1 x} }\) | \(=\) | \(\ds \lim_{x \mathop \to 0} \paren {\frac 1 x - \map \Gamma x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{x \mathop \to 0} \paren {\frac 1 x - \frac {\map \Gamma {x + 1} } x}\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{x \mathop \to 0} \paren {\frac {\map \Gamma 1 - \map \Gamma {1 + x} } x}\) | $\map \Gamma 1 = 1$ as Gamma Function Extends Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds -\map {\Gamma'} 1\) | Definition of Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \gamma\) | Derivative of Gamma Function at 1 |
$\blacksquare$
Historical Note
François Le Lionnais and Jean Brette, in their Les Nombres Remarquables of $1983$, attribute this to a $1976$ result of K. Demys, but this has not been corroborated.
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,57721 56649 \ldots$