Have you heard about $\gamma$ ?

Calculus Level 5

$\large \dfrac1\gamma \lim _{ x\rightarrow 0 }{ \ln { \sqrt [ x ]{ x! } } } =\, ?$

Notation : $\gamma$ denotes the Euler-Mascheroni constant , $\gamma \approx 0.5772$ .

The answer is -1.

1 solution

Subhadeep Biswas
Jun 14, 2016

$\dfrac{1}{\gamma} \lim_{x\to0}\ln\sqrt[x]{x!}$ $=\dfrac{1}{\gamma} \lim_{x\to0}\dfrac{\ln x!}{x}$ $=\dfrac{1}{\gamma} \lim_{x\to0}\dfrac{\ln \Gamma(x+1)}{x}$ $=\dfrac{1}{\gamma} \lim_{x\to0}\dfrac{\dfrac{1}{\Gamma(x+1)}\times \Gamma(x+1)\times\psi(x+1)}{1} \text{[ using L'Hospital's rule]}$ $=\dfrac{1}{\gamma} \lim_{x\to0} \psi(x+1)$ $=\dfrac{1}{\gamma} \psi(1)$ $=\dfrac{1}{\gamma}\times(-\gamma)$ $=-1$

We don't need de l'Hopital here. $\lim_{x\to0}\frac{\ln \Gamma(x+1)}{x}=\psi(1)$ by the first principles definition of differentiation of $\ln\Gamma(x+1)$ at $x=0$ .

Mark Hennings - 5 years ago

Yes Sir, I didn't notice that, thank you for pointing it ^ ^

Subhadeep Biswas - 5 years ago

Have you heard about γ \gamma γ ?

The answer is -1.

1 solution

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Have you heard about $\gamma$ ?