Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/35

31

DES ÉQUATIONS

${\begin{array}{rll}&+\left({\frac {1}{2^{21}}}.{\frac {1}{41}}+{\frac {1}{2^{21}}}.{\frac {1}{42}}+{\frac {1}{2^{22}}}.{\frac {1}{43}}\right)&+{\frac {1}{2^{20}}}.{\frac {2215}{41.42.43}}\\&-\left({\frac {1}{2^{23}}}.{\frac {1}{45}}+{\frac {1}{2^{23}}}.{\frac {1}{46}}+{\frac {1}{2^{24}}}.{\frac {1}{47}}\right)&-{\frac {1}{2^{22}}}.{\frac {2656}{45.46.47}}\\&+\left({\frac {1}{2^{25}}}.{\frac {1}{49}}+{\frac {1}{2^{25}}}.{\frac {1}{50}}+{\frac {1}{2^{26}}}.{\frac {1}{51}}\right)&+{\frac {1}{2^{24}}}.{\frac {3137}{49.50.51}}\\&-\ldots \ldots \ldots \ldots \ldots \ldots &-\ldots \ldots \ldots \ldots \end{array}}$

série dont le terme général est

$\left(-{\frac {1}{4}}\right)^{n-1}.{\frac {20n^{2}-19n+4}{(4n-3)(4n-2)(4n-1)}}.$

En réduisant ses termes en décimales, on aura

${\begin{array}{rll}{\frac {\varpi }{4}}&=+{\frac {5}{1.2.3}}&=+0{,}833333333\\&-{\frac {1}{4^{1}}}.{\frac {46}{5.6.7}}&=\ldots \ldots \ldots \ldots \ldots -0{,}054761905\\&+{\frac {1}{4^{2}}}.{\frac {127}{9.10.11}}&=+0{,}008017677\\&-{\frac {1}{4^{3}}}.{\frac {248}{13.14.15}}&=\ldots \ldots \ldots \ldots \ldots -0{,}001419414\\&+{\frac {1}{4^{4}}}.{\frac {409}{17.18.19}}&=+0{,}000274795\\&-{\frac {1}{4^{5}}}.{\frac {610}{21.22.23}}&=\ldots \ldots \ldots \ldots \ldots -0{,}000056061\\\end{array}}$