Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/34

30

INTÉGRATION

${\frac {\varpi }{4}}=0{,}785398163397.$

Pour y parvenir, il ne s’agira que de faire $x=1,$ dans la formule ci-dessus ; les termes, dont l’indice est divisible par $4,$ disparaîtront d’eux-mêmes, et il viendra

${\begin{array}{rll}{\frac {\varpi }{4}}=&+\left({\frac {1}{2}}.{\frac {1}{2}}\ \ \ +{\frac {1}{2}}.{\frac {1}{2}}\quad +{\frac {1}{2^{2}}}.{\frac {1}{3}}\right)=&+{\frac {5}{1.2.3}}\\&-\left({\frac {1}{2^{3}}}.{\frac {1}{5}}\ \ +{\frac {1}{2^{3}}}.{\frac {1}{6}}\ \ +{\frac {1}{2^{4}}}.{\frac {1}{7}}\right)&-{\frac {1}{2^{2}}}.{\frac {46}{5.6.7}}\\&+\left({\frac {1}{2^{5}}}.{\frac {1}{9}}\ \ +{\frac {1}{2^{5}}}.{\frac {1}{10}}\ +{\frac {1}{2^{6}}}.{\frac {1}{11}}\right)&+{\frac {1}{2^{4}}}.{\frac {127}{9.10.11}}\\&-\left({\frac {1}{2^{7}}}.{\frac {1}{13}}\ +{\frac {1}{2^{7}}}.{\frac {1}{14}}\ +{\frac {1}{2^{8}}}.{\frac {1}{15}}\right)&-{\frac {1}{2^{6}}}.{\frac {248}{13.14.15}}\\&+\left({\frac {1}{2^{9}}}.{\frac {1}{17}}\ +{\frac {1}{2^{9}}}.{\frac {1}{18}}\ +{\frac {1}{2^{10}}}.{\frac {1}{19}}\right)&+{\frac {1}{2^{8}}}.{\frac {409}{17.18.19}}\\&-\left({\frac {1}{2^{11}}}.{\frac {1}{21}}+{\frac {1}{2^{11}}}.{\frac {1}{22}}+{\frac {1}{2^{12}}}.{\frac {1}{23}}\right)&-{\frac {1}{2^{10}}}.{\frac {610}{21.22.23}}\\&+\left({\frac {1}{2^{13}}}.{\frac {1}{25}}+{\frac {1}{2^{13}}}.{\frac {1}{26}}+{\frac {1}{2^{14}}}.{\frac {1}{27}}\right)&+{\frac {1}{2^{12}}}.{\frac {851}{25.26.27}}\\&-\left({\frac {1}{2^{15}}}.{\frac {1}{29}}+{\frac {1}{2^{15}}}.{\frac {1}{30}}+{\frac {1}{2^{16}}}.{\frac {1}{31}}\right)&-{\frac {1}{2^{14}}}.{\frac {1132}{29.30.31}}\\&+\left({\frac {1}{2^{17}}}.{\frac {1}{33}}+{\frac {1}{2^{17}}}.{\frac {1}{34}}+{\frac {1}{2^{18}}}.{\frac {1}{35}}\right)&+{\frac {1}{2^{16}}}.{\frac {1453}{33.34.35}}\\&-\left({\frac {1}{2^{19}}}.{\frac {1}{37}}+{\frac {1}{2^{19}}}.{\frac {1}{38}}+{\frac {1}{2^{20}}}.{\frac {1}{39}}\right)&-{\frac {1}{2^{18}}}.{\frac {1814}{37.38.39}}\\\end{array}}$