Exercice 6-bis NiM+SSdEaWdpdHNHNiIiI0k= restart:Digits:=30: NiM+SSJmRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUknYXJjdGFuRzYkSSpwcm90ZWN0ZWRHRi9JKF9zeXNsaWJHRiU2IzkkRiVGJUYl NiM+SSJhRzYiIiIh NiM+SSJiRzYiIiIi NiM+SSJORzYiIiQrIg== restart:Digits:=30:N:=100: Grace \340 l'in\351galit\351 de Taylor-Lagrange abs(f(b) -Sum(Diff(f(a),t$k)/(k!)*(b-a)^k,k=0..'N'))<=sup(abs(Diff(f(t),t$(k+1))))/((n+1)!)*(b-a)^(n+1); NiMxLUkkYWJzR0kqcHJvdGVjdGVkR0YmNiMsJi1JImZHNiI2I0kiYkdGKyIiIi1JJFN1bUc2JEYmSShfc3lzbGliR0YrNiQqKC1JJURpZmZHRjE2JC1GKjYjSSJhR0YrLUkiJEdGJjYkSSJ0R0YrSSJrR0YrRi4tSSpmYWN0b3JpYWxHRiY2I0Y/ISIiKSwmRi1GLkY6RkNGP0YuL0Y/OyIiIUkiTkdGK0ZDKigtSSRzdXBHRis2Iy1GJTYjLUY2NiQtRio2I0Y+LUY8NiRGPiwmRj9GLkYuRi5GLi1GQTYjLCZJIm5HRitGLkYuRi5GQylGRUZZRi4= On etudie donc la suite de Taylor (Taylor_n) associ\351e \340 la fonction t->4/(1+t^2) Taylor[n]:=Sum(Diff(f(a),t$k)/(k!)*(b-a)^k,k=0..'N')=Taylor[n-1]+Diff(f(a),t$k)/(n!)*(b-a)^n; NiM+JkknVGF5bG9yRzYiNiNJIm5HRiYvLUkkU3VtRzYkSSpwcm90ZWN0ZWRHRi1JKF9zeXNsaWJHRiY2JCooLUklRGlmZkdGLDYkLUkiZkdGJjYjSSJhR0YmLUkiJEdGLTYkSSJ0R0YmSSJrR0YmIiIiLUkqZmFjdG9yaWFsR0YtNiNGPCEiIiksJkkiYkdGJkY9RjdGQUY8Rj0vRjw7IiIhSSJOR0YmLCYmRiU2IywmRihGPUZBRj1GPSooRjFGPS1GP0YnRkEpRkNGKEY9Rj0= Taylor:=proc(f,a,b,N) options remember;
local S;
if N=0 then S:=f(a); else
S:=((D@@N)(f))(a)*(b-a)^N/N!+Taylor(f,a,b,N-1); fi;
S;
end: Appliquons ceci \340 la fonction arctan entre 0 et 1 pour N=100 tels que ci-dessousf:=t->arctan(t);a:=0;b:=1;N:=100; for i from 0 to N do 4*Sum(((D@@k)(f))(a)*(b-a)^k/k!,k=0..i)=4*Taylor(f,a,b,i), erreur=evalf(4*Taylor(f,a,b,i)-Pi); od; NiQvLCQtSSRTdW1HNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYkKiYtLS1JI0BAR0YnNiRJIkRHRidJImtHRio2I0kiZkdGKjYjIiIhIiIiLUkqZmFjdG9yaWFsR0YoNiNGMyEiIi9GMztGN0Y3IiIlRjcvSSdlcnJldXJHRiokIT9HJFFWRVlRS3oqZWBFZlRKISNI NiQvLCQtSSRTdW1HNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYkKiYtLS1JI0BAR0YnNiRJIkRHRidJImtHRio2I0kiZkdGKjYjIiIhIiIiLUkqZmFjdG9yaWFsR0YoNiNGMyEiIi9GMztGN0Y4IiIlRj8vSSdlcnJldXJHRiokIj5zO21OUDp3MS1UWXRTZSkhI0g= NiQvLCQtSSRTdW1HNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYkKiYtLS1JI0BAR0YnNiRJIkRHRidJImtHRio2I0kiZkdGKjYjIiIhIiIiLUkqZmFjdG9yaWFsR0YoNiNGMyEiIi9GMztGNyIiIyIiJUZAL0knZXJyZXVyR0YqJCI+czttTlA6dzEtVFl0U2UpISNI 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