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Prove-that-the-Euler-Constant-is-qlso-equal-to-lim-x-1-x-1-x-x-1-




Question Number 117826 by snipers237 last updated on 13/Oct/20
  Prove that the Euler Constant is qlso equal to  lim_(x→−1)   Γ(x)−(1/(x(x+1)))
$$\:\:{Prove}\:{that}\:{the}\:{Euler}\:{Constant}\:{is}\:{qlso}\:{equal}\:{to} \\ $$$$\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\:\:\Gamma\left({x}\right)−\frac{\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)} \\ $$
Answered by mnjuly1970 last updated on 14/Oct/20
solution::   lim_(x→−1)  [Γ(x)−(1/(x(x+1)))]   =lim_(x→−1 )  (((Γ(x+2)−1)/(x(x+1))))   =^(l′hospital) lim_(x→−1 ) (((Γ ′(x+2))/(2x+1)))        =((Γ′(1))/(−1))=−Γ′(1)=−ψ(1)Γ(1)=γ   where::  γ  is Euler−  Masheroni constant           ... m.n.1970...
$${solution}:: \\ $$$$\:{lim}_{{x}\rightarrow−\mathrm{1}} \:\left[\Gamma\left({x}\right)−\frac{\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)}\right] \\ $$$$\:={lim}_{{x}\rightarrow−\mathrm{1}\:} \:\left(\frac{\Gamma\left({x}+\mathrm{2}\right)−\mathrm{1}}{{x}\left({x}+\mathrm{1}\right)}\right) \\ $$$$\:\overset{{l}'{hospital}} {=}{lim}_{{x}\rightarrow−\mathrm{1}\:} \left(\frac{\Gamma\:'\left({x}+\mathrm{2}\right)}{\mathrm{2}{x}+\mathrm{1}}\right) \\ $$$$\:\:\:\: \\ $$$$=\frac{\Gamma'\left(\mathrm{1}\right)}{−\mathrm{1}}=−\Gamma'\left(\mathrm{1}\right)=−\psi\left(\mathrm{1}\right)\Gamma\left(\mathrm{1}\right)=\gamma \\ $$$$\:{where}::\:\:\gamma\:\:{is}\:\mathscr{E}{uler}− \\ $$$$\mathscr{M}{asheroni}\:{constant} \\ $$$$\:\:\:\:\:\:\:\:\:…\:{m}.{n}.\mathrm{1970}… \\ $$

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