[PDF] 16°) Les suites un+1 = f(u les calculs que tu peux

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???? ??? ?????? ?un+1=f(un)?? ??u0? f=x7!x2u ?? ??? ??????? ????f? n 1 n+1 1 =a b 0 1 :u n 1 a b 0 1 ??D=a0 0 1 P=1b 0 1a M n=1b 0 1a :an0 0 1 :1 0 b1a :11a? ?? ??????? ?? ????? ??????x7!a:x+b? ?????b1a? ?? ??????? ?vn+1=un+1b1a=a:un+bb1a=a:una:b1a=a:vn?????? ?? ??????? ?????vn=an:v0 ?? ??????? ?un=vn+b1a=an: u 0b1a u

2=a:(a:u0+b) +b=a2:u0+a:b+b

u

3=a:(a2:u0+a:b+b) =a3:u0+a2:b+a:b+b

u

4=a:(a3:u0+a2:b+a:b+b) +b=a4:u0+a":b+a2:b+a:b+b

? ??????? ??? ?? ??? ? ???? ???? ? ??????u ?? ?? ?????u0?? ?? ??????un+1=a:un+bc:u n+d c d 03 1 1 1 2 1 1 1 2 :2 0 03 :21 1 1 =75 108
u n+1=7:un510:un8 108
=1 1 1 2 :2 0 03 :21 1 1 75
108
n =1 1 1 2 :2n0

0 (3)n

:21 1 1 u ???? ??????? ????? ??????? ????n= 0?u0=1:u0+ 00:u0+ 1????? ??? ??n= 1?u1=7:u0510:u08 ???? ?????? ???? ??????? ?? ?????? ??un? ??????? ? u n=(2:rn1):u0+ 1rn2:(rn1):u0+ 2rn????r=23 u n!u0+ 12:u0+ 2=12 ?? ???? ?? ??????? ???12 ??? ???? ??? ???? ?????? ???? ??f? 12 n12 u n1?=unfixeattu nfixerep??P 1=21

1 1?? ??????? ????? ????? ??? ?????? ?

n+1=7:un510:un812 n=P1:M:P u ?? ? ?????? ????? ?? ????? ?? ???? ?????(a6un6b)):::)(a6un+16b)?? ???? ??????? ?? ?? ????? ? ?????? ????? ???? ?????? ??? ??? ?????? ?????? ?x2[a; b])f(x)2[a; b]?

8x2I; f(x)2I

?? ?? ???????up? ????? ? ????I????? ???? ???uk????k???? ????? ??? p??????? ????I? ?? ????? ?? ???? ??? ?????? ???? ?a6f(x)6b? f(b) n+1=p2:un+ 3u n+1=2:un+ (un)24 h 32
;3i [3;+1[??? ??????[2;0]??? ?????? [0;2]??? ?????? [2;+1[??? ?????? u n+1=un(un)2u n+1=(un)323 ] 1;0]??? ?????? [0;1]??? ?????? [1;+1[????? ??? ??????? ????[1;+1[??????? ???]1;0]??? ??? ???????] 1;1]??? ??????? [1;2]??? ??????? [2;+1[??? ??????? ???? ?? ??? ????? ?????? ??? ????? ?? ???? ?????[0;2] u n+1=p2:un+ 3?u 0i 1;32 hi 32
3? ?????(un)?????? ?? ?????u1??????? ???? ?? ????75 ????u2????45 ?? ??????? ?8x2I; f(x)6x ????? ?? ?????(un)??? ? ?????? ????I

8n; f(un)6un?? ????8n; un+16un?

?? ??????? ?8x2I; f(x)>x ????? ?? ?????(un)??? ? ?????? ????I ????? ?? ???? ??b? ???? ????? ?? ????? ?? ??????? ???? ????? ?? ????? ? ?????? ???? ??????? ?? ????? ??un+1un? 1;32 hh 32
???? ?? ????? ??? ?u+1=f(un)???????=f()?? i 1;32 hh 32
32
n%3un!3u h 32
;+1h u n+1=2:un+ (un)24

X2+ 2:X4

X???? ?X:(X2)4

?u n&02u ???? ??? ???? ?? ?????? ? ?? ???? ??? ????? ?? ????? ???? ?? ??? ????? ????0??2? ?? ????? ??? ? ??????? ????]2;0[???? ?? ???? ???? ????? ???? ???? ?????? ???u0??? ???? ???? ????? ???2? ??? ?? ??????? ?? ??? ??????? ??? ?? ????? ??????? ????+1? ? ? ????? ??????? ??????? ????+1??

0] 1;4[4]4;2[2u

1=f(u0)u

>>22u

12]0;2[0

u n+1=un(un)2 u n? [0;1]0 i;1 +i:::)? ???????2? z ?? ?? ???? ????? ?? ?????z0=a+i:b? u n+1=(un)323 ?? ??? ??? ?????? ???????1??2? ] 1;1[1 f(un)< un)? ?? ?? ????u

0< u1u

????un+1< un+2? u ??? ?? ????? ????un+1= cos(un)? f=x7!x+a? (Pn+1; Vn+1; Tn+1) =fonction de(pn; Vn; Tn)? x

0dt1 +t26Z

x 0 dt=x? ? ????? ??1n

2? ?????

1ln(n)? ????? ??2n?? ????? ????? ???????

?? ??????dn=1(un+1)21(un)2? ????? ????? ?????? ???? ??????? ???? ??? ?????? ?? ?? ?????u???? u n???? ????0?

1(un+1)21(un)2=1(Arctan(un))21(un)2=(unArctan(un)):(Arctan(un) +un)(un)2:(Arctan(un))2?

???? ?? ?? ???????(xArctan(x)):(x+Arctan(x))(x:Arctan(x))2?????x???? ????0? ?? ??? ?????? ??? ???????

23
d0+d1+:::+dn1n

1(un)21(u0)2n

???? ????23 pn:u 2 unq 3 u nr3 2:n 1pn u u u ?????x???? ????0?

Arctan

0(0)? +o(x3)x!0? ?? ????? ??Arctan(x) =Arctan(0)+x:Arctan0(0)+x22 :Arctan"(0)+x36 :Arctan(3)(0)+x46 :Z 1 0 (1t)3:Arctan(4)(t:x):dt? xArctan(x) =x33 +o(x3)x!0x!0x33 33
:2:x(x:x)2? z n+1=zn+jznj2 ???? ?? ???? ?????? ?????? ?? ????? ?jzn+1j6jzn+jznjj2

6jznj?

?? ????? ???? ???(u2:n)??????u2:(n+1)=f(f(u2:n))? =n:1 +ei:n2

1 +ei:2

1 +ei:2

=ei:=2:ei:=2+ei:=22 ?? ? ????zn+1=n:1 +ei:n2 =n:cos(n=2):ei:n=2? n+1=n:cosn2 n+1=n2 ????0? n=0:cos02 :cos04 :::cos02 n n ?? ??????? ????? ?????? ?? ???? ????? ???? ?? ???????cos():sin() =sin(2:)2 cos 02 n :sin02 n =12 :sin02 n1 ?? ???? ?n:sin02 n =0:cos02 :cos04 :::cos02 n :sin02 n =0:12 n:sin()??? ? ? ??n n=0:sin(0)2 n:sin02 n ?? ?? ?????? ???0 sin(x)x !x!01?? ??????sin(x)x!0x? ?? ? ????n=0:sin(0)2 n:sin02 n n!+10:sin(0)2 n:92 n n!+10:sin(0) 0? n!0 n!0:sin(a) 0z n!0:sin(0)

19352006????? ?? ???

?? ??? ???? ????? ??????? ?? ? ? ??? ??????? ??z0???? f(un; un1) '(an; bn) (an)??? ??? ????? ?????? ? ??????? ????[0;1]? ??(un)??? ?????? ???u0????? ?? ??? ????n? u n+1=11 +an+un? ?? ? ?????un+1=11 ++un11 ++?????? ????? ???? u n+1=un(1 ++):(1 ++un)??? ?????? ?????? u u n+11un? 1? ????un+1=11 +an+un?? ?? ?????? ???an???? ??????? ?? ?? ????? ?? ?? ????? ??? ? ?? ??????

1? ? ??? ??? ?? ?????

n 13 ????0? u n+1bn+1=11 +an+un11 ++bn=(an) + (bnun)(1 +an+un):(1 +bn+)6(an) + (bnun)(1 +)? ?? ??????("n)?? ?????(unbn)? ?? ? ??????j"n+1j6j"nj+janj1 +? ?? ??? ?? ?????? ?j"nj6j"0j(1 +)n+n1X k=0janj(1 +)nk? n1X 0? ????? ?? ?? ???????f0= 0)f constante??f0>0)f croissante? ??????? ???? ??????? ???f(b)?? ???? ??????? ???f(a)?

4f1?????? ???? ??[f(a); f(b)]???[a; b]

?? ?????y??y0????[f(a); f(b]????y < y0 ?? ?????? ?????f1(y)< f1(x0)?????[a; b]? f(f1(y))>f(f1(y0))?? ????y>y0?

8" >0;9

">0;9">0;8y; (y0 "6y6y0+"))(f1(y0)"6f1(y)6 f

1(y0) +")

f

1(y0)"6f1(y)6f1(y0) +"

ff1(y0)"6y6f f

1(y0) +"

?? ?? ???? ?????y0 "=ff1(y0)"??y0+"=ff1(y0) +" "=y0ff1(y0)"?? "=ff1(y0) +"y0 ?? ?? ?????? ??? ?y6y0+",y6y0+ff1(y0) +"y0 y6y0+",y6ff1(y0) +"??? ?????? y6y0+",f1(y)6f1 ff1(y0) +" f 1 ????R????? ????R????? [f(a); f(b] ??? ??? ?????? ?????? ??R2? f(a)? f() ??????? ?exp??(R;+)????[R+;)?

8(a;)2A2; f(a+) =f(a)+f()

8(a;)2A2; f(a) =f(a)?f()

??????? ?z7!z??(C;+;:)???? ??? ?????

8(!u ;!v)2A2; f(!u+!v) =f(!u)+

f(!v)

8!u2A;82K; f(:!u) =

:f(!u) ?? ???? ???x7!Z x 0 "=y0ff1(y0)"??"=ff1(y0) +"y0? ?(jxaj60))(x=a)) jf(x)f(a)j6"? ???? ??????(jxaj61))(everything you want!) f

1(y0)?? ????ff1(y0)"< f(f1(y0)) =y0????y0ff1(y0)">0?

?? ????? ??? ??????? ?? ??????'=t7!t3+ 2:t? y t= 1cos(t) ???? ????1cos? ?? ?? ?? ????? ??? ??????? ??? Z x

0p1 + (f0(x))2:dx?

=f7!Z b as1 + (f0(t))22:g:f(t):dt? f(b) =B? ?? ???? ???? ? ??????? ??? ??????? ?f?? ????? ? ???? ???????(f+df)(f)????df??? ??????? ?? ?????? ??f? ????? ? ??? ?9??f(a+h) =f(a) +A:h+o(h)h!0f(a+h)f(a)h f(a+h)f(a)h =A+o(1)???? ??? ???????f(a+h)f(a) =quotesdbs_dbs47.pdfusesText_47

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