Exercice 6 restart:Digits:=30:N:=100: Grace \340 l'in\351galit\351 de Taylor-Lagrange abs(f(1) -Sum(Diff(f(0),t$k)/(k!),k=0..'N'))<=sup(abs(Diff(f(t),t$(k+1))))/((n+1)!); NiMxLUkkYWJzR0kqcHJvdGVjdGVkR0YmNiMsJi1JImZHNiI2IyIiIkYtLUkkU3VtRzYkRiZJKF9zeXNsaWJHRis2JComLUklRGlmZkdGMDYkLUYqNiMiIiEtSSIkR0YmNiRJInRHRitJImtHRitGLS1JKmZhY3RvcmlhbEdGJjYjRj4hIiIvRj47RjlJIk5HRitGQiomLUkkc3VwR0YrNiMtRiU2Iy1GNTYkLUYqNiNGPS1GOzYkRj0sJkY+Ri1GLUYtRi0tRkA2IywmSSJuR0YrRi1GLUYtRkI= On etudie donc la suite de Taylor (Taylor_n) associ\351e \340 la fonction t->4/(1+t^2) Taylor[n]:=4*Sum((-1)^k/(2*k+1),k=0..n)=Taylor[n-1]+4*(-1)^n/(2*n+1); NiM+JkknVGF5bG9yRzYiNiNJIm5HRiYvLCQtSSRTdW1HNiRJKnByb3RlY3RlZEdGLkkoX3N5c2xpYkdGJjYkKiYpISIiSSJrR0YmIiIiLCZGNCIiI0Y1RjVGMy9GNDsiIiFGKCIiJSwmJkYlNiMsJkYoRjVGM0Y1RjUqJilGM0YoRjUsJkYoRjdGNUY1RjNGOw== Taylor:=proc(N) options remember;
local S;
if N=0 then S:=4; else
S:=4*(-1)^N/(2*N+1)+Taylor(N-1); fi;
S;
end: for i from 0 to N do 4*Sum((-1)^k/(2*k+1),k=0..i)=evalf(Taylor(i)), erreur=evalf(abs(Taylor(i)-Pi)); od; NiQvLCQtSSRTdW1HNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYkKiYpISIiSSJrR0YqIiIiLCZGLyIiI0YwRjBGLi9GLzsiIiFGNSIiJSRGNkY1L0knZXJyZXVyR0YqJCI+czttTlA6dzEtVFl0U2UpISNI NiQvLCQtSSRTdW1HNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYkKiYpISIiSSJrR0YqIiIiLCZGLyIiI0YwRjBGLi9GLzsiIiFGMCIiJSQiP25tbW1tbW1tbW1tbW1tRSEjSC9JJ2VycmV1ckdGKiQiPmg7bihmenJsN0JwKWYjXFpGOQ== NiQvLCQtSSRTdW1HNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYkKiYpISIiSSJrR0YqIiIiLCZGLyIiI0YwRjBGLi9GLzsiIiFGMiIiJSQiP25tbW1tbW1tbW1tbW1tTSEjSC9JJ2VycmV1ckdGKiQiPlIkR0JTP0dNKG8yOFMyRCRGOQ== NiQvLCQtSSRTdW1HNiRJKnByb3RlY3RlZEdGKEkoX3N5c2xpYkc2IjYkKiYpISIiSSJrR0YqIiIiLCZGLyIiI0YwRjBGLi9GLzsiIiEiIiQiIiUkIj9DJjRRXzRRXzRRXzRRXypHISNIL0knZXJyZXVyR0YqJCI+LylHMHVPKyEpcF4kZVhOWSNGOg== 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