SESSION 05
> restart: series(1/(1+x),x=0,5);
1 - x + x2 - x3 + x4 + O( )x5
> series(1/(1+x),x=1,4);
1 2 - 1
4 x - 1 + 1 8(x - 1)2 - 1
16(x - 1)3 + O((x - 1)4)
> series(1/(1-x^4),x,10);
1 + x4 + x8 + O(x12)
> Order;
6
> series(log(1+x)-x-x^2/2,x=0);
-x2 + 1 3 x3 - 1
4 x4 + 1
5 x5 + O( )x6
> Order:=10;
Order := 10
> series(log(1+x)-x-x^2/2,x=0);
-x2 + 1 3 x3 - 1
4 x4 + 1 5 x5 - 1
6 x6 + 1 7 x7 - 1
8 x8 + 1
9 x9 + O(x10)
> Order:=6;
Order := 6
> series(exp(x),x);
1 + x + 1 2 x2 + 1
6 x3 + 1 24 x4 + 1
120 x5 + O( )x6
> series(sin(x),x);
x - 1 6 x3 + 1
120 x5 + O( )x6
> series(cos(x),x);
1 - 1 2 x2 + 1
24 x4 + O( )x6
> series(sinh(x),x);
x + 1 6 x3 + 1
120 x5 + O( )x6
> series(cosh(x),x);
1 + 1 2 x2 + 1
24 x4 + O( )x6
> series((1+x)^a,x);
1 + a x + 1
2 a (a - 1) x2 + 1
6 a (a - 1) (a - 2) x3 + 1
24 a (a - 1) (a - 2) (a - 3) x4 + 1
120 a (a - 1) (a - 2) (a - 3) (a - 4) x5 + O( )x6
> series(1/(1-x),x);
1 + x + x2 + x3 + x4 + x5 + O( )x6
> f:=x-> cos(x)-x; plot(f(x),x=-1..1); plot(f(x),x=0.739..0.7395);
f := x ! cos( ) - xx
> a:=series(f(x),x=Pi/4,2); p:=convert(a,polynom); solve(p); evalf(%);
plot([f(x),p],x=-1..1);
a := 1 2 2 - 1
4 !
"
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' + - 1 2 2 - 1
"
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' x - 1
4 !
"
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' + O x - 1
4 !
"
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'
" 2
##
$
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&
' p := 1
2 2 - 1 4 ! + - 1
2 2 - 1
"
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' x - 1
4 !
"
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' 2 4 + !( )
4 2 + 2( ) 0.7395361338
> a:=series(f(x),x=Pi/4,3); p:=convert(a,polynom): solve(p); evalf(%);
plot([f(x),p],x=-1..1);
a := 1 2 2 - 1
4 !
"
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' + - 1 2 2 - 1
"
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' x - 1
4 !
"
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' - 1
4 2 x - 1 4 !
"
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' 2
+ O x - 1 4 !
"
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'
" 3
##
$
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&
' - 1
64 2 32 2 + 64 - 8 2 ! + 32 10 + 4 2 - 2 !( ), - 1
64 2 32 2 + 64 - 8 2 ! - 32 10 + 4 2 - 2 !( ) -3.996722841, 0.7390920425
> restart: p:=sin(x): u:=series(p,x,10); q:=convert(u,polynom);
plot([p,q],x=-5..5); plot(p-q,x=-100..100);
u := x - 1 6 x3 + 1
120 x5 - 1
5040 x7 + 1
362880 x9 + O(x10) q := x - 1
6 x3 + 1 120 x5 - 1
5040 x7 + 1 362880 x9
> restart: p:=sin(x):
plot([p,seq(convert(series(p,x,n),polynom),n=2..20)],x=-10..10, y=-3..3);
> restart: f:=x->tan(tanh(x))-tanh(tan(x)); series(f(x),x,8);
series(f(x),x,12);
f := x ! tan tanh( ( )x) - tanh tan( ( )x) 2
45 x7 + O( )x8 2
45 x7 - 26
4725 x11 + O(x12)
> f:=x->1/(1+x^2): plot(f(x),x=-10..10);
u:=series(f(x),x,7); p:=convert(u,polynom); plot([f(x),p],x=-2..2, y=-2..2);
u := 1 - x2 + x4 - x6 + O( )x8 p := 1 - x2 + x4 - x6
> L:=[]: for i from 1 to 18 do L:=[op(L),convert(series(f(x),x,2*i+1), polynom)] od: L;
plot([f(x),seq(L[k],k=1..8)],x=-2..2,y=-2..2);
plot([f(x),seq(L[k],k=8..16)],x=-2..2,y=-2..2);
[1 - x2, 1 - x2 + x4, 1 - x2 + x4 - x6, 1 - x2 + x4 - x6 + x8, 1 - x2 + x4 - x6 + x8 - x10, 1 - x2 + x4 - x6 + x8 - x10 + x12, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14,
1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20,
1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20 - x22, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20 - x22 + x24, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20 - x22 + x24 - x26, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20 - x22 + x24 - x26 + x28, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20 - x22 + x24 - x26 + x28 - x30, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20 - x22 + x24 - x26 + x28 - x30 + x32, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20 - x22 + x24 - x26 + x28 - x30 + x32 - x34, 1 - x2 + x4 - x6 + x8 - x10 + x12 - x14 + x16 - x18 + x20 - x22 + x24 - x26 + x28 - x30 + x32 - x34 + x36 ]
En incluant des ordres de plus en plus élevés, l'expression est de mieux en mieux approximé dans l'intervalle de -1 à 1, mais au voisinage des extrémités de l'intervalle le développement limité tend vers une grandeur très grande positive ou négative. Le rayon de convergence de la série semble être de 1.
> restart: f:=x->exp(-x): plot(f(x),x=0..10);
u:=series(f(x),x,7); p:=convert(u,polynom); plot([f(x),p],x=0..10, y=0..2);
u := 1 - x + 1 2 x2 - 1
6 x3 + 1 24 x4 - 1
120 x5 + 1
720 x6 + O( )x7 p := 1 - x + 1
2 x2 - 1 6 x3 + 1
24 x4 - 1 120 x5 + 1
720 x6
> L:=[]: for i from 1 to 36 do L:=[op(L),convert(series(f(x),x,i), polynom)] od:
plot([f(x),seq(L[k],k=1..18)],x=0..10,y=0..2);
plot([f(x),seq(L[k],k=18..36)],x=0..10,y=0..2);
On voit graphiquement que l'exponentielle est bien approximée par un développement à l'ordre n sur un intervalle de -n a +n. Sans que cela soit une démonstration, on peut supposer que le rayon de convergence est infini, ce qui se démontre proprement par ailleurs.
> restart: u:=series(sin(x),x): v:=series(cos(x),x): series(u*v,x);
series(u/v,x); series(tan(x),x);
series(u+v,x);
x - 2 3 x3 + 2
15 x5 + O( )x6 x + 1
3 x3 + 2
15 x5 + O( )x6 x + 1
3 x3 + 2
15 x5 + O( )x6 1 + x - 1
2 x2 - 1 6 x3 + 1
24 x4 + 1
120 x5 + O( )x6
> restart: u:=series(tan(x),x,7); series(ln(1+u),x,7);
u := x + 1 3 x3 + 2
15 x5 + O( )x7
x - 1 2 x2 + 2
3 x3 - 7 12 x4 + 2
3 x5 - 31
45 x6 + O( )x7
> u:=series(1/(1+x),x=0,7); int(u,x);
u := 1 - x + x2 - x3 + x4 - x5 + x6 + O( )x7 x - 1
2 x2 + 1 3 x3 - 1
4 x4 + 1 5 x5 - 1
6 x6 + 1
7 x7 + O( )x8
> v:=series(ln(1+x),x,8); diff(v,x);
v := x - 1 2 x2 + 1
3 x3 - 1 4 x4 + 1
5 x5 - 1 6 x6 + 1
7 x7 + O( )x8 1 - x + x2 - x3 + x4 - x5 + x6 + O( )x7
> u:=series(tan(x),x); solve(u=y,x); series(arctan(x),x);
u := x + 1 3 x3 + 2
15 x5 + O( )x7 y - 1
3 y3 + 1
5 y5 + O( )y6 x - 1
3 x3 + 1
5 x5 + O( )x6
> restart: f:=x->(1+x)*tan(Pi/(x+4));
plot(f(x),x=-10..10, y=-20..20,discont=true);
u:=series(f(x),x); re:=solve(u=y,x);
v:=convert(re,polynom); v:=subs(y=x,v);
plot([f(x),v,x],x=-3..3,y=-3..3,discont=true,color=[pink,blue,green]);
f := x ! (1 + x) tan ! x + 4
"
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' u := 1 + - 1
8 ! + 1
"
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' x + - 3 32 ! + 1
128 !2
"
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' x2 + 3
128 ! + 1 256 !2 - 1
1536 !3
"
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' x3
+ - 3 512 ! - 5
2048 !2 - 1
6144 !3 + 5 98304 !4
"
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' x4 + 3
2048 ! + 1 1024 !2 + 1
4096 !3 - 1 245760 !5
"
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' x5 +
O( )x6 re := - 8
! - 8(y - 1) + 4 ! -12 + !( )
! - 8 ( )3
y - 1
( )2 - 4 ! -576 + !( 3 + 408 ! - 44 !2) 3 (! - 8)5
y - 1 ( )3
- 4 ! -804 !( 3 + 29 !4 + 6640 !2 - 15744 ! + 9216) 3 (! - 8)7
y - 1 ( )4
+ 2 ! 165344 !( 4 + 3 !7 - 4546560 ! + 68 !6 + 3886080 !2 - 8204 !5 + 1474560 - 1275840 !3) 15 (! - 8)9
y - 1
( )5 + O((y - 1)6) v := - 8 (y - 1)
! - 8 + 4 ! -12 + !( ) (y - 1)2
! - 8 ( )3
- 4 ! -576 + !( 3 + 408 ! - 44 !2) (y - 1)3 3 (! - 8)5
- 4 ! -804 !( 3 + 29 !4 + 6640 !2 - 15744 ! + 9216) (y - 1)4 3 (! - 8)7
+
2 ! 165344 !( 4 + 3 !7 - 4546560 ! + 68 !6 + 3886080 !2 - 8204 !5 + 1474560 - 1275840 !3) (y - 1)5 15 (! - 8)9
v := - 8 (x - 1)
! - 8 + 4 ! -12 + !( ) (x - 1)2
! - 8 ( )3
- 4 ! -576 + !( 3 + 408 ! - 44 !2) (x - 1)3 3 (! - 8)5
- 4 ! -804 !( 3 + 29 !4 + 6640 !2 - 15744 ! + 9216) (x - 1)4 3 (! - 8)7
+
2 ! 165344 !( 4 + 3 !7 - 4546560 ! + 68 !6 + 3886080 !2 - 8204 !5 + 1474560 - 1275840 !3) (x - 1)5 15 (! - 8)9
> restart: f:=x->sinh(x) - ((a*x+b*x^3+c*x^5)/(1 + d*x^2+e*x^4));
u:=series(f(x),x,11); p:=convert(u,polynom); co:=coeffs(p,x);
so:=[solve({co})]; subs(so[1],f(x)); u:=series(subs(so[1],f(x)),x,12);
f := x ! sinh( ) - x a x + b x3 + c x5 1 + d x2 + e x4
u := 1 - a( ) x + 1 6 - b + a d
"
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' x3 + -c + a e - -b + a d( ) d + 1 120
"
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' x5
+ - -b + a d( ) e - -c + a e + d b - a d( 2) d + 1 5040
"
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' x7
+ 1
362880 - -c + a e + d b - a d( 2) e - (e b - 2 e a d + d c - d2 b + a d3) d
"
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' x9 + O(x11)
p := 1 - a( ) x + 1 6 - b + a d
"
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' x3 + -c + a e - -b + a d( ) d + 1 120
"
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' x5
+ - -b + a d( ) e - -c + a e + d b - a d( 2) d + 1 5040
"
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' x7
+ 1
362880 - -c + a e + d b - a d( 2) e - (e b - 2 e a d + d c - d2 b + a d3) d
"
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' x9
co := 1 - a, 1
6 - b + a d, -c + a e - -b + a d( ) d + 1
120, - -b + a d( ) e - -c + a e + d b - a d( 2) d + 1 5040, 1
362880 - -c + a e + d b - a d( 2) e - (e b - 2 e a d + d c - d2 b + a d3) d so := c = 551
166320, a = 1, e = 5 11088, d = -13
396, b = 53 396 ()
*
+, - ./
0
12 3 sinh( ) - x
x + 53
396 x3 + 551 166320 x5 1 - 13
396 x2 + 5 11088 x4 u := 11
457228800 x11 + O(x12)
>
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