[PDF] [PDF] Exercices de Colles de Sup

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?? ??????? ???A[B=A[C??A\B=A\C? ??????? ???B=C?

P(F)?????? ???(A) =fhAi?

? ??????? ???f(A\B)f(A)\f(B)? ? ???????E=N?E???? ???? ???????g(n) =n+1?f(n) =n????n1?? f(0) = 1? ????E? ????n2N?f(n+1m ??E??? ?????? ????x2E??A=ffn(x)jn2Ng? ??A6=E? ????? ???? ?????? ??? ????? ??? ??AB?f(A)f(B)? ??????? ???f????? ?? ????? ???? k=0k(k+1)!? k+1???n k? ?? ? ?? ??????? ??p?? n= 2p?? ??p??p+ 1??n= 2p+ 1? 8< :x

3y4z= 2

xyz 4= 4 x

5yz2= 16

k=1k2 k? ?????? ??? ?? ?????? ?????n! 1? ???qk? ?? ??????212 n1n2 n? ???? ?? ?????? ????2? X

A;BEjA\Bj

?? ?????? ?? ??????? ?? ????? ???A?????jBj2n1? m+n k =X p+q=k m p n q pii1p i1? ??d(n) = (1+

1):::(k+ 1)

k=11p k???? ????+1?????n! 1? k=1111p k???? ????1? ?? P n1 +:::+1n ? ?? ????? ?? ????? ???????ln 111p
k ?? ?? ?????a(1 +i)??a(1i)????a2R? ?? ?? ?????? ???? ?????? ???? ?? ??????0?3p2?1+i3 p2 ??1i3 p2 zuz1u2R???juj= 1? k=0cos3k? 34
cot2n14 cot32n a+ib=c+id? ??????? ??? ??? ?????? ??????a?b?c??d??????? ?? ?????? f(z) =ziz+i?

01? ?? ??????? ???f1(z0)2H???

jz0j<1? X k1[3] k n

2n+(n)3

????(n) =1;1;2;1;1;2????? ?? ?????? ??n??????6? 2nX k=0cos 2n +k2n n2 2n12n n? ? ?? ?mb=!(ma)???? ?? ? ??? ????? ??m? ??????? ????? ??????1;:::;k2 f1;1g???? ???j1z1+:::+kzkj p3? arctan(x) + arctan(2x) =4 4 4 = 4arctan15 arctan1239

0< beaxaebx< ba

??????? ?x? Z 1 0 xklnxdx 0??? coshx+ coshy=a sinhx+ sinhy=b e e x? ?? ???? ?? ?? ???? ???a >0??a2b2+ 4?

R??? ?x???????ln1+axln1+bx?

Z sin2x 0 arcsinptdt

2+x+1?

2+1? ?? ?? ??? ?????? ???

2x+1p3

lnjcosxj x22 ;2 2

0lnsinxdx?

2 2 x? ?? ?I=J? ??I+J=2 ln2? +910
cost310 sint? x0=xy y 0=x+y ????y(0) = 0??x(0) = 1? ??y??? ?????y(t) =et(Acost+Bsint)??A= 0? ????x(t) =Betcost???? z=x+iy???

8x2R;f0(x) =f(ax)?

????f(x) =Asin(x+4 a2 )????A2R? lntan t2 +4 ? ?? ??????y= 2arctanecosx+2 (mod)? ? ?? ?????? ?????? ??? ????a=2ikT y

00+y= max(ex;1)

12 (excosxsinx))? ?? ?? ????? ???? ???? ??????? ?? ??????? ?? ?? ???? ????? ?? ???????y0?? ?? ?????? ???y00(0) = 1? ? ??y(t)t???? ????t? ?? ??????y(t) =t1Cet????C0? ? ??y(t)t???? ????t? ?? ??????y(t) =t+ 1 +Cet????C0? ? ??y(t)t???tt0? ????? ????tt0?y(t) =t1+et0t??? ????tt0? y(t) =t+ 1ett0? ?? ?? ?????? ??? ?? ??????? ??? ????C1? ???f??? ??????? f()0? f ??tn? t n+1tn+1tnt ????tn! 1?tn+1tn??tnp2n???t2n+1t2n!2? sup x;y2Ajxyj= supAinfA x >supA2 ??y2A??? ???y ??? ???jun+1unj< ????nN? ?? ????? ?? ??????mN??? ???jumxj< ??n > m??? ???junyj< ? ?? ?????? ?????k?????m??n??? ???jukzj< ? ??l >1? ????? ????nN?(n)> n? ???? ????? ??? ?????? ?????N?? ???? u ????u???? ????0????u3n=o(nun)???? (nun)???? ????1? ?? ???? ????un=1n vn?1n vn

3=nvn????nvn1n

3 ??un=1n 1n 4+o1n 4? ? ????u??? ????? ?????? ????? ??? ???? ????m??n??N?um+num+un? ??????? ???unn ????? ???? ??????? ???(c1n ???um(+)m? ???? ?????n=km+r????r < m? u km+rkm+r++maxr2[[0;m1]]urkm + 2 ????k????? ?????? ????unn + 2????n????? ?????? ????unn ????feg??G? ????Z? ??????n??? ???zpn= 1? p x 2=e? ?2jHj? ??? ?? ?????? ???H[xH? G? ??????? ???AB=G? x????G??? ???x2=e? ???G??? ???? ???A??? ?? ?????? ???A??? ?? ?????? ? ????x2A????x6= 0?x22x2A????x2x2A????x=x2a????a2A? ?? ????? ???? ????k?2kA??? ?? ????? ??A? ?? ?????? ??? ?? ???? ??? ????? ? ??I??? ?? ?????? ?? ?????2kpq ????? ??y!ya??? ?????? ?? ?? ??? ???? ????k?????? ?ank??? ?????? ???? ????n? ???? ?? ??????p < q???? ???apk=aqk? ????xqpk=k? ?? ??????? xk= 06=k?

1 + 1 + 2 + 3 +:::+n=n(n+ 1)2

+ 1 n 3? a(1106)???? ?? ???? ?? ???????1061 mod 13? modn2? ???????p1 mod 24 ??(600;30)? n n(n1)2 +1 ??????? ??E? a

1+a2+:::+ak=n?

S n;k=Sn;k1+Sn1;k1+:::+S0;k1???? ?? ???? ?? ??????? ??? ??????? ?? E? ? ??????? ???Dn+1=n(Dn+Dn1)???? ????n1? ? ?? ??????? ???Dn=nDn1+(1)n? ???? ??????Dn???? ?? ????? ????? D n=n!12! 13! +:::+(1)nn! a b=ba? d baa0b=b0a????a0= 1? ???? ?? ???? ??????b=ka? ????aka=ba??ak=ak? ???? ????a >0?(2;4)??(4;2)? ??(4k0;0)???? ????k0??????? n X i=11i ??????n??k???? ???A\[[n;+1[[=kN\[[n;+1[[? ?? ???k= 1? ?? ?k=PGCD(a1;a2;:::;am)?? ???ai???? ????A? ???? i=1uiai= 1? ????A f(x) x?f(2x) =f(x)? ???? ????x? ????f= 0? ??????? ???f????? ?? ?????? ????? ???? h(t) = sup [a;b]f+tg g(x) = sup [x;x+1]f gf? ??????? ????? ??????x??? ???f(x) =g(x)?

A??B????R???? ??? ???? ????x?jf(x)j Ajxj+B?

jf(0)j+n??? ???jxj n?

2????x6= 0? ??f(0) = 0?

??? ?Z'(x) x et2 ???? ??????? ???'???C1? ?????? ??????? ???x? P 0? ??????? ???f??? ?????? ????? ??f(x) "? ???jf(n)(x)j jP(x)j???? ????n??x? ??????? ???f??? ?????? ?????? ??0? ???? ????x? ?? ?????? ????c?????0??x??? ???f(x) =cn+1(n+1)!f(n+1)(c)? ??????? ???f0??? ?????? ???2pM 0M2? 2 M0h +hM22 0M 2? g

0(b) = 0

(1 + sinx)1x +x36 x412 +x524 +o(x5) (1 + sinx)1x =e(112 x+724 x2316 x3+1391152 x4+o(x4)) rx tanx +215
x5+o(x6) xtanx= 1x23 x445 +o(x5)px tanx= 1x26 x440 +o(x5) x xxxx (x1)3+o((x1)3) x xx=x+ (x1)2+32 (x1)3+o((x1)3) ? ?????? ?? ?? ??g??0? ???????6? y3+11120 y5+o(y6)?? ?? ln2n2n2+o(ln2nn 2) +ln2ln222n2 rx

1+:::+xnn

!!0npx

1:::xn

[??un+1= sinun???? ????n?? N ? ????unq3 n ??f(0) =f0(0) =f00(0) = 0? ?? ?????? ?????f0(x)22f(x)M(x)? ???? ??? ?? ??????? ??P??? ??????? ??hmin=f0(x)M(x)? ?? ?? ? ????jhminj jxj??? 4? ???R??? ???? ????z??C? ?? ?jP(z)j jIm(z)jn? ?? ????P= (Xa1):::(Xan)1? X n1jP2X? ??? ?????P(X)P(X)?n??????? ???? ??? ??? ????P2R[X]? ????? ??????P1??P2????R[X]???? ???P=P21+P22? ?? ????Q=P+P0+:::+P(n)? ??????? ???Q(x)0???? ????x2R? ????? ?? ??????? ??x0? ?? ?? ??????Q(x0)0? n X k=1kxk1 n(n+1)2 k2[[0;n]]? ??????? ????P(n+ 1)? ??0;1;:::;d? ????x2Ed??y2R? ?? ?f(x;y) =Pd i=0f(x;i)Li(y)? ????? X

3(X2+ 1)2

2+1X(X2+1)2

=2Z? ???? ????p2N? ?????? ?? ???????p????? ?? ? 1X

22Xcos+ 1

2isin(1)pp!(1(Xei)p+11(Xei)p+1)

n=11n(n+1)(n+2)

1n(n+1)(n+2)=12n1n+1+12(n+2)? ???? ?? ????? ????14

F n+Gn1Fn1+:::+G1F+G0= 0 ??????? ??? ?????? ???? ??F??? ???? ???? ????? ?? ???Gi? (Xa)n? F=n1X k=0! kX!k F???X!k?? ?? ???? ??????X????!k? ?? ?? ??????Xn1??!k? ?? ??????F=nx n1? Pn i=11x iP0(xi)=1P(0)? ? ??????? ????Pn i=11P

0(xi)?

n= 1?0?????? +1Q ??????F=P(X)n??G=+1Q ? ????FG=QnP(+1Q X i=1P

00(xi)P

0(xi) P0P ???? ???????0? ? ??????n2N??z1;:::;zn??? ??????? ??Xn+ 1? ??????? ??? ???? ????

P??Cn[X]? ?? ? ?

XP

0(X) =n2

P(X) +2n

n X k=1z kP(zkX)(zk1)2 ? ??P2C[X]? ?? ????kPk= supjzj=1jP(z)j? ??????? ??? ???? ????P2C[X]? ?? ?kP0k nkPk? F=XlX jzkjj(zk1)2j? ??? ?? ??????zk=ei? ?? ?????? ???zk(zk1)22R? ???? ????? ???? ?????n24 ??????? ??E??f2L(E;F)? ??????? ???f(V)f(W)???V+Ker(f)W+Ker(f)? ????x??E? ?? ???????(x;f(x))??? ????? fg=Id?

E????Im(p) =Im(q)?

???p??q? ???? =2 f0;1g? ????a;b2C? ????? ?????? ?? ??????g2L(E)??? ???fg=Id? ??????? ???f??? ?? gf=Id? ????E????Im(p)Ker(q)? ??r=p+qpq? g2Gg1pg? ?? ?????? ??? ???? ???? ????i2[[0;k]]?fi2L(Ei;Ei+1)? ?? ??????? ??? ???? ????i2 [[0;k1]]?Ker(fi+1) =Im(fi)? kX i=1(1)idim(Ei) = 0 ??????? ???rg(vu) =rg(u)???Im(u)\Ker(v) =f0g? ??????? ???rg(u) +rg(v) =n? uvu=u? R ??0??? ?? ????? ???niai??? ????? ????? ?? ??????? ??????n????+1? ??ai ????? ???dim(U)? ?? ??????? ?? ??????? ????En(U[V)? C

0(R;R)?

L?? ???? ???K????

??????KLM????? ?????? ??????? ??? ? [M:K] = [M:L][L:K] ???L? ?? ?? ?????? ??? ???eifj??????? ??? ???? ??M???K? ???? ???????0????? ?? ????n?

0? ?? ?????? ???(x;u(x);:::;uq1(x))??? ??????

rg(u3) +rg(u)2rg(u2) f(U+V) =f(U) +f(V)f(U[V) fg2I?? ??????f??g???? ???Ker(fu+gv) =Ker(u)\Ker(v)? ?? ???? ????V=T A k=In+nk1n1J? ??????? ??????? ??? ??? ??ji < k 2)

2? ???? ??? ?? ???? ??A?

????? ?? ???? ?? ??????g??? ??? ???? ??Im(f)? ?????? ??Mn(R)??? ?M???????AMB? jai;ij>X i6=jjai;jj ???? ????i;j2[[1;n]]??k2N? ?? ????a(k) (k) j= minia(k) i;j?(k) j= maxia(k) i;j??(k) j=(k) j(k) j? ????? ????= mini;jai;j? ? ???????(k) j(k+1) j(k+1) j(k)quotesdbs_dbs46.pdfusesText_46

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