Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/271

 Les corrections sont expliquées en page de discussion

${\begin{array}{r|r}&1\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\1&1+\ {\frac {1}{2}}x+{\frac {1.3x^{2}}{2.4}}+\quad {\frac {1.3.5x^{3}}{2.4.6}}+\qquad {\frac {1.3.5.7x^{4}}{2.4.6.8}}+\ldots ,\\\\-{\frac {2}{x}}&-\ \ {\frac {1}{2}}x-{\frac {1.3x^{2}}{2.4}}-\quad {\frac {1.3.5x^{3}}{2.4.6}}-\qquad {\frac {1.3.5.7x^{4}}{2.4.6.8}}+\ldots ,\\\\2&+{\frac {1.2}{2.4}}x-\ {\frac {12x^{2}}{2.4.6}}-\quad {\frac {90x^{3}}{2.4.6.8}}-\qquad {\frac {840x^{4}}{2.4.6.8.10}}+\ldots ,\\\\-{\frac {2}{x}}&+\ \ {\frac {6x^{2}}{2.4.6}}+\quad {\frac {60x^{3}}{2.4.6.8}}+\qquad {\frac {630x^{4}}{2.4.6.8.10}}+\ldots ,\\\\2&+{\frac {24x^{2}}{2.4.6.8}}+\quad {\frac {360x^{3}}{2.4.6.8}}+\quad {\frac {5040x^{4}}{2.4.6.8.10.12}}+\ldots ,\\\\-\quad {\frac {2}{x}}&-{\frac {120x^{3}}{2.4.6.8.10}}-\quad {\frac {2520x^{4}}{2.4.6.8.10.12}}-\ldots ,\\\\2&-{\frac {720x^{3}}{2.4.6.8.10.12}}-{\frac {20160x^{4}}{2.4.6.8.10.12.14}}-\ldots ,\\\\-{\frac {2}{x}}&+{\frac {5040x^{4}}{2.4.6.8.10.12.14}}+\ldots ,\\\\\ldots &\ldots \ldots \,;\end{array}}$

d’où résultera

${\frac {1}{\sqrt {1-x}}}={\cfrac {1}{1+{\cfrac {1}{-{\cfrac {2}{x}}+{\cfrac {1}{2+{\cfrac {1}{-{\cfrac {2}{x}}+{\cfrac {1}{2+{\cfrac {1}{-{\cfrac {2}{x}}+{\cfrac {1}{2+\ldots }}}}}}}}}}}}}},$

ou bien encore