47
DES FRACTIONS CONTINUES.
![{\displaystyle C=-{\frac {2}{3!}},\quad C'=+{\frac {4}{5!}},\quad C''=-{\frac {6}{7!}},\quad C'''=+{\frac {8}{9!}},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3874c92cc2dbe7edb0c628c71a16b04930a6999a)
![{\displaystyle D=+{\frac {2.8}{3!5!}},\quad D'=-{\frac {4.12}{3!7!}},\quad D''=+{\frac {6.16}{3!9!}},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc0934369e22db003a20e30c23c59ea08225369)
![{\displaystyle E=+{\frac {2.8.48}{3!5!7!}},\quad E'=-{\frac {4.12.64}{3!5!9!}},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6f9e5ab125fe97d3f457d163ffd085744ebadb8)
![{\displaystyle F=-{\frac {2.8.48.128}{3!5!7!9!}},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bed05c435c8c152149daedea8ce3366fb104f62)
Nous avons donc finalement
![{\displaystyle A=+1,\ B=+1,\ C=-{\tfrac {1}{3}},\ D=+{\tfrac {1}{45}},\ E=+{\tfrac {1}{4725}},\ F=-{\tfrac {1}{13395375}},\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2654e10d60dde86ee91e0898eeb1ac61bbc645)
puis donc qu’on doit avoir
![{\displaystyle {\frac {\operatorname {Tang} .x}{x}}={\cfrac {B}{A+{\cfrac {Cx^{2}}{B+{\cfrac {Dx^{2}}{C+{\cfrac {Ex^{2}}{D+{\cfrac {Fx^{2}}{E+\ldots }}}}}}}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36cb92327ad6e2f723ab449124c6465d11fe2656)
on aura
![{\displaystyle {\frac {\operatorname {Tang} .x}{x}}={\cfrac {1}{1-{\cfrac {{\cfrac {1}{3}}x^{2}}{1+{\cfrac {{\cfrac {1}{45}}x^{2}}{-{\cfrac {1}{3}}+{\cfrac {{\cfrac {1}{4725}}x^{2}}{{\cfrac {1}{45}}-{\cfrac {{\cfrac {1}{13395375}}x^{2}}{{\cfrac {1}{4725}}+\ldots }}}}}}}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99a5a071aca9fbec13a7787b6c0d76153eb55afd)
ce qui donne, en amenant successivement les numérateurs à être
entiers négatifs, et en multipliant ensuite par ![{\displaystyle x^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1a303560dc80c8b384c576fcc2a8ff7dedc1c5)
![{\displaystyle x\operatorname {Tang} .x={\cfrac {x^{2}}{1-{\cfrac {x^{2}}{3-{\cfrac {x^{2}}{5-{\cfrac {x^{2}}{7-{\cfrac {x^{2}}{9-{\cfrac {x^{2}}{11-\ldots }}}}}}}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c964edd2a89c46bd7bb7cf9a6935956b8743abff)
résultat dont la loi est manifeste, et qui, quel que soit
satis-