[PDF] [PDF] Chapitre14 : Développements limités - Melusine

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I a I

f:IÑRnPN

ε:IÑR 0a

@xPI,f(x) =λ0+λ1(x´a) +λ2(x´a)2+¨¨¨+λn(x´a)n+ (x´a)nε(x) f(x) =λ0+λ1(x´a) +λ2(x´a)2+¨¨¨+λn(x´a)n+o((x´a)n)

λ0,λ1,...λnPR

f(a+u) =λ0+λ1u+λ2u2+¨¨¨+λnun+o(un) 0 f:uÞÑf(a+u) 2 4 4 +x) =? 2 2 (x´x) =? 2 2 (1´x2 2 +o(x2)´(x+o(x2))) 2 2 (1´x´x2 2 +o(x2)) e x+x3x= (1 +x+x2 2! +x3 3! +o(x3)) +o(x3) = 1 +x+x2 2! +x3 3! +o(x3) %e ´1 x

6x‰0

0x= 0 f(x) = 0 +o(x5) =o(x5)

0,λ1,...λn

0,µ1,...µn 0

f(a+u) =λ0+uλ1+¨¨¨+unλn+o(un) f(a+u) =µ0+uµ1+¨¨¨+unµn+o(un) @iPJ0,nK,λi=µi

ε,η 0

@uPJ,λ0+λ1u+¨¨¨+λnun+unε(u) =µ0+µ1u+¨¨¨+µnun+unη(u)

ɍJ=tuPR,a+uPIu

u= 0

λ0=µ0

u

@uPJzt0u,λ1+¨¨¨+λnun´1+un´1ε(u) =µ1+¨¨¨+µnun´1+un´1η(u)

u0 λ1=µ1Ę @uPJzt0u,λn´1+λnu+uε(u) =µn´1+µnu+uη(u)

λn´1=µn´1

@uPJzt0u,λn+ε(u) =µn+η(u) λn=µn 'p=n păn

f(a+u) =λ0+λ1u+¨¨¨+λpup+λp+1up+1+¨¨¨+λnun+unε(u)looooooooooooooooooooomooooooooooooooooooooon

Ñ0 =o(up) f(a+u) =f(a) +ε(u) ɍε 00 f a f(a+u)´f(a)ÝÝÝÑuÑ00 f(a+u) =f(a) +o(1) a f(a) =λ0 f(a+u) =f(a) +uf1(a) +o(u) u= 0λ0=f(a) u‰0f(a+u)´f(a) u f:RÝÑR xÞÝÑ$ %x 31
x x‰0 0x= 0 f(x) =o(x2) 0 x‰0f1(x) = 3x21 x

´x1

x f1(x)´f1(0) x´0= 3x1 x looomooon

Ñ0´1

x loomoon nPN f(a+u) =f(a) +f1(a)ˆu+f(2)(a) 2! u2+¨¨¨+f(n)(a) n!un+o(un) f:RÝÑR xÞÝÑx3+x f RR C

8 f´1(0),(f´1)1(0),(f´1)2(0),(f´1)3(0)

f

´1(x) =a0+a1x+a2x2+a3x3+x3ε(x)

@kPJ0,3K,ak=(f´1)(k)(0) k!. x f f ´1(f(x)) =a0+a1(x+x3) +a2(x+x3)2+a3(x+x3)3+ (x+x3)3ε(x+x3) f ´1(f(x)) =a0+a1x+a1x3+a2x2+o(x3) +a3x3+o(x3) +x3(1 +x2)3loooomoooon

Ñ1ε(x+x3)loooomoooon

Ñ0looooooooooooomooooooooooooon

=o(x3) f´1(f(x)) =a0+a1x+a2x2+ (a1+a3)x3+o(x3) a0= 0a1= 1a2= 0a3=´1 f´1(0) = 0(f´1)1(0) = 1(f´1)2(0) = 0(f´1)3(0) = 3!ˆ(´1) =´6 0 n0 uPR e

1+u=eˆeu=e+eu+eu2

2 +eu3 6 +eu3ε(u)

ɍε(u)ÝÝÝÑuÑ00

f,g:IÑRλPR

λff+g f

λ fg

f(a+x) =a0+a1x+¨¨¨+anxnlooooooooooooomooooooooooooon

P(x)+xnε(x)

g(a+x) =b0+b1x+¨¨¨+bnxnloooooooooooomoooooooooooon

Q(x)+xnη(x)

(f(a+x) +g(a+x) =P(x) +Q(x) +xnη(x)loomoon =o(xn)+xnε(x)loomoon =o(xn) e x= 1 +x+x2 2! +x3 3! +o(x3) x= 1´x2 2 +o(x3) ex+x= 2 +x+x3 6 +o(x3) ex+ 2x= 3 +x´x2 2 +x3 6 +o(x3)

1 +x= 1 +1

2 x´1 8 x2+1 16 x3+o(x3) x=x´x3 6 +o(x3) x?

1 +x= (x´x3

6 +o(x3))(1 +1 2 x´1 8 x2+1 16 x3+o(x3)) =x´x3 6 +1 2 x2´1 8 x3+o(x3) =x+x2 2

´7x3

24
+o(x3) fg f(a+x) =a0+a1x+¨¨¨+anxnlooooooooooooomooooooooooooon

P(x)+xnε(x)

g(a+x) =b0+b1x+¨¨¨+bnxnloooooooooooomoooooooooooon

Q(x)+xnη(x)

f(a+x)g(a+x) =P(x)Q(x) +P(x)xnη(x)loooooomoooooon o(xn)+Q(x)xnε(x)loooooomoooooon =o(x2n)=o(xn) e x= 1 +x+x2 2! +¨¨¨+xn n!+xnε(x)

ɍε(x)ÝÝÝÑxÑ00

e

´x= 1´x+x2

2! +¨¨¨+ (´1)nxn n!+xn(´1)nε(´x)loooooomoooooon Ñ0 f:IÑRaPI g:JÑRɍJ f(I)ĂJ f(a+x) =λ0loomoon g(λ0+u) =µ0+µ1u+µ2u2+¨¨¨+µnun+unη(u),ɍη(u)ÝÝÝÑuÑ00

g(f(a+x)) =g(λ0+λ1x+λ2x2+¨¨¨+λnxn+xnε(x)loooooooooooooooooooooomoooooooooooooooooooooon

u)

+ (λ1x+λ2x2+¨¨¨+λnxn+xnε(x))nloooooooooooooooooooooooomoooooooooooooooooooooooon

x

nˆ λn1xÑ0η(λ1x+λ2x2+¨¨¨+λnxn+xnε(x))loooooooooooooooooooooooomoooooooooooooooooooooooon

=o(xn) x x= 1´x2 2 +o(x3) x=c

1´x2

2 +o(x3) =?

1 +uu=´x2

2 +o(x3) uÝÝÝÑxÑ00

1 +u= 1 +u

2

´u2

8 +u3 16 +u3η(u) x= 1´x2 4 +o(x3)´1 8 (´x2 2 +o(x3))2+1 16 (´x2 2 +o(x3))3 + (´x2 2 +o(x3))3η(´x2 2 +o(x3))loooooooomoooooooon =o(x6) = 1´x2 4 +o(x3)

1 +u= 1 +u

2

´u2

8 +u2η(u) x=x´x3 6 (x)3=x3(1´x2 6 x=x+o(x2) x= 1´x2 2 +x4 4! +o(x4) e x=(1´x2 2 +x4 4! +o(x4)) =e(´x2 2quotesdbs_dbs33.pdfusesText_39

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