exo6-7-8-9.mw

Exercice 6

> f:=x->arctan(x^2):f(x)=series(f(x),x=0,7);

arctan(x^2) = (series(x^2-1/3*x^6+O(x^7),x,7))

> f:=x->sin(2*x)-2*(exp(x)-1):f(x)=series(f(x),x=0,8);

sin(2*x)-2*exp(x)+2 = (series(-x^2-5/3*x^3-1/12*x^4+1/4*x^5-1/360*x^6-13/504*x^7+O(x^8),x,8))

> f:=x->sin(x)*exp(cos(x)):f(x)=series(f(x),x=0,3);

sin(x)*exp(cos(x)) = (series(exp(1)*x+O(x^3),x,3))

> f:=x->x^2/sin(x)^2:f(x)=series(f(x),x=0,6);

x^2/sin(x)^2 = (series(1+1/3*x^2+1/15*x^4+O(x^6),x,6))

> f:=x->sin(x)*sinh(x):f(x)=series(f(x),x=0,6);

sin(x)*sinh(x) = (series(x^2+O(x^6),x,6))

> f:=x->ln(1+x*cos(x)):f(x)=series(f(x),x=0,5);

ln(1+x*cos(x)) = (series(x-1/2*x^2-1/6*x^3+1/4*x^4+O(x^5),x,5))

> f:=x->x*sin(x)^2/(1+x+x^2):f(x)=series(f(x),x=0,7);

x*sin(x)^2/(1+x+x^2) = (series(x^3-x^4-1/3*x^5+4/3*x^6+O(x^7),x,7))

> f:=x->ln(exp(x)+exp(-x)):f(x)=series(f(x),x=0,7);

ln(exp(x)+exp(-x)) = (series(ln(2)+1/2*x^2-1/12*x^4+1/45*x^6+O(x^8),x,8))

>

> Exercice 7

> f:=x->(ln(1+x))^2:f(x)=series(f(x),x=0,5);

ln(1+x)^2 = (series(x^2-x^3+11/12*x^4+O(x^5),x,5))

> f:=x->ln(1+x^2)*exp(x):f(x)=series(f(x),x=0,5);

ln(1+x^2)*exp(x) = (series(x^2+x^3+O(x^5),x,5))

> f:=x->ln(1+x^2)*exp(x^2):f(x)=series(f(x),x=0,7);

ln(1+x^2)*exp(x^2) = (series(x^2+1/2*x^4+1/3*x^6+O(x^8),x,8))

> f:=x->ln(cos(x)):f(x)=series(f(x),x=0,7);

ln(cos(x)) = (series(-1/2*x^2-1/12*x^4-1/45*x^6+O(x^8),x,8))

> f:=x->sin(tan(x)):f(x)=series(f(x),x=0,5);

sin(tan(x)) = (series(x+1/6*x^3+O(x^5),x,5))

> f:=x->exp(sin(x)):f(x)=series(f(x),x=0,4);

exp(sin(x)) = (series(1+x+1/2*x^2+O(x^4),x,4))

> f:=x->sin(x)^6:f(x)=series(f(x),x=0,10);

sin(x)^6 = (series(x^6-x^8+O(x^10),x,10))

>

> Exercice 8

> f:=x->(ln(1+x))*(sin(x)+cos(x)):f(x)=series(f(x),x=0,4);

ln(1+x)*(sin(x)+cos(x)) = (series(x+1/2*x^2-2/3*x^3+O(x^4),x,4))

> f:=x->(sqrt(1+x)-1)^2*(1-cos(x)):f(x)=series(f(x),x=0,8);

((1+x)^(1/2)-1)^2*(1-cos(x)) = (series(1/8*x^4-1/16*x^5+11/384*x^6-17/768*x^7+O(x^8),x,8))

> f:=x->ln(sin(x)+cos(x)):f(x)=series(f(x),x=0,4);

ln(sin(x)+cos(x)) = (series(x-x^2+2/3*x^3+O(x^4),x,4))

> f:=x->exp(cos(x)):f(x)=series(f(x),x=0,5);

exp(cos(x)) = (series(exp(1)-1/2*exp(1)*x^2+1/6*exp(1)*x^4+O(x^5),x,5))

> f:=x->(cos(x))^(1/x):f(x)=series(f(x),x=0,3);

cos(x)^(1/x) = (series(1-1/2*x+O(x^2),x,2))

> f:=x->sin(Pi*cos(x)):f(x)=series(f(x),x=Pi/3,3);

sin(Pi*cos(x)) = (series(1-3/8*Pi^2*(x-1/3*Pi)^2+O((x-1/3*Pi)^3),x = 1/3*Pi,3))

>

> Exercice 9

> f:=x->1/sin(1/x):f(x)=series(f(x),x=infinity,6);

1/sin(1/x) = x+1/6/x+7/360/x^3+O(1/x^4)

> f:=x->exp(1/x)/((exp(1/x)-1)*ln((1+x)/x)):f(x)=series(f(x),x=infinity,3);

exp(1/x)/((exp(1/x)-1)*ln((1+x)/x)) = x^2+x+1/4+1/24/x-1/72/x^2+O(1/x^3)