Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/276

d’où on conclura le développement que voici

${\cfrac {e^{x}+1}{e^{x}-1}}={\cfrac {2}{x}}+{\cfrac {1}{{\cfrac {6}{x}}+{\cfrac {1}{{\cfrac {10}{x}}+{\cfrac {1}{{\cfrac {14}{x}}+{\cfrac {1}{{\cfrac {18}{x}}+{\cfrac {1}{\cfrac {22}{x+\ldots }}}}}}}}}}}\,;$

et, par suite

${\cfrac {e^{x}-1}{e^{x}+1}}={\cfrac {1}{{\cfrac {2}{x}}+{\cfrac {1}{{\cfrac {6}{x}}+{\cfrac {1}{{\cfrac {10}{x}}+{\cfrac {1}{{\cfrac {14}{x}}+{\cfrac {1}{{\cfrac {18}{x}}+{\cfrac {1}{\cfrac {22}{x+\ldots }}}}}}}}}}}}}\,;$

ou bien encore

$x.{\cfrac {e^{x}-1}{e^{x}+1}}={\cfrac {x^{2}}{2+{\cfrac {x^{2}}{6+{\cfrac {x^{2}}{10+{\cfrac {x^{2}}{14+{\cfrac {x^{2}}{18+{\cfrac {x^{2}}{22+\ldots }}}}}}}}}}}}$

En posant x=1, on tire de là

${\cfrac {e-1}{e+1}}={\cfrac {1}{2+{\cfrac {1}{6+{\cfrac {1}{10+{\cfrac {1}{14+{\cfrac {1}{18+{\cfrac {1}{22+\ldots }}}}}}}}}}}}\,;$

ce qui donne les réduites successives

${\cfrac {1}{2}},\quad {\frac {6}{13}},\quad {\frac {61}{132}},\quad {\frac {860}{1861}},\quad {\frac {15541}{33630}},\quad {\frac {342762}{741721}},\ldots$