> t1:=series(sin(x),x=0, 3);

t1 := series(1*x+O(x^3),x,3)

> t2:=series(sin(x),x=0,5);

t2 := series(1*x-1/6*x^3+O(x^5),x,5)

> t3:=series(sin(x),x=0,7);

t3 := series(1*x-1/6*x^3+1/120*x^5+O(x^7),x,7)

> t4:=series(sin(x),x=0,9);

t4 := series(1*x-1/6*x^3+1/120*x^5-1/5040*x^7+O(x^9...

> t5:=series(sin(x),x=0,21);

t5 := series(1*x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362...
t5 := series(1*x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362...

> P1:=convert(t1,polynom);

P1 := x

> P2:=convert(t2,polynom);

P2 := x-1/6*x^3

> P3:=convert(t3,polynom);

P3 := x-1/6*x^3+1/120*x^5

> P4:=convert(t4,polynom);

P4 := x-1/6*x^3+1/120*x^5-1/5040*x^7

> P5:=convert(t5,polynom);

P5 := x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362880*x^9-1...
P5 := x-1/6*x^3+1/120*x^5-1/5040*x^7+1/362880*x^9-1...

> plot({sin(x), P1},x=-4..4, title=`sin(x) and Taylor polynomial T_1`);

[Maple Plot]

> plot({sin(x), P2},x=-4..4, title=`sin(x) and Taylor polynomial T_3`);

[Maple Plot]

> plot({sin(x), P3},x=-4..4, title=`sin(x) and Taylor polynomial T_5`);

[Maple Plot]

> plot({sin(x), P4},x=-4..4, title=`sin(x) and Taylor polynomial T_7`);

[Maple Plot]

> plot({sin(x), P5},x=-4..4, title=`sin(x) and Taylor polynomial T_19`);

[Maple Plot]

>