Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/279

${\begin{array}{r|l}{\frac {1}{x}}&{\frac {x}{1}}-{\frac {x^{2}}{2}}\ \ +{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}\ +{\frac {x^{5}}{5}}-{\frac {x^{6}}{6}}+\ldots \\\\{\frac {2}{1}}&{\frac {x}{2}}-{\frac {x^{2}}{2}}\ \ +{\frac {x^{3}}{4}}-{\frac {x^{4}}{5}}\ +{\frac {x^{5}}{6}}-{\frac {x^{6}}{7}}+\ldots \\\\{\frac {3}{x}}&\qquad {\frac {x^{2}}{6}}\ \ -{\frac {x^{3}}{7}}+{\frac {3x^{4}}{20}}-{\frac {2x^{5}}{15}}+{\frac {5x^{6}}{42}}-\ldots \\\\{\frac {2}{2}}&\qquad {\frac {x^{2}}{6}}\ \ -{\frac {x^{3}}{5}}+{\frac {x^{4}}{5}}\ -{\frac {4x^{5}}{21}}+{\frac {5x^{6}}{28}}-\ldots \\\\{\frac {5}{x}}&\qquad \qquad +{\frac {x^{3}}{30}}-{\frac {x^{4}}{20}}\ +{\frac {2x^{5}}{35}}-{\frac {5x^{6}}{84}}+\ldots \\\\{\frac {2}{3}}&\qquad \qquad +{\frac {x^{3}}{20}}-{\frac {3x^{4}}{35}}+{\frac {9x^{5}}{84}}-\ldots \\\\{\frac {7}{x}}&\qquad \qquad \qquad +{\frac {x^{4}}{140}}\ -{\frac {x^{5}}{70}}+\ldots \\\\\ldots &\qquad \qquad \qquad +{\frac {x^{4}}{70}}-\ldots \end{array}}$

ce qui donnera

$\operatorname {Log} .(1+x)={\cfrac {1}{{\cfrac {1}{x}}+{\cfrac {1}{{\cfrac {2}{1}}+{\cfrac {1}{{\cfrac {3}{x}}+{\cfrac {1}{{\cfrac {2}{2}}+{\cfrac {1}{{\cfrac {5}{x}}+{\cfrac {1}{{\cfrac {2}{3}}+{\cfrac {1}{{\cfrac {7}{x}}+{\cfrac {1}{{\cfrac {2}{4}}+\ldots }}}}}}}}}}}}}}}}\,;$

ou bien encore