[PDF] [http://mp.cpgedupuydelome.fr] édité le 17 août 2017

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S n=nX k=0cos(k)??un=cos(n)n ??? ??????? ?? ????? (Sn)n2N??? ??????? ?? ?????? ?????? cos(n) =SnSn1? ??????? ??? ?? ????? ?? ????? ??????? u u n= cosn2ln(11=n)? ????x2]0;1[[]1;+1[? ?? ???? f(x) =Z x2 xdtlnt? ??????? ??? ? ??????? x >1? Z x2 xxdttlntf(x)Z x2 xx

2dttlnt?

?? ??????? ?? ?????? ??f??1+ ??????? ?? ???? ?? ?????? ?? f??1? f 0(x)? ??????? Z+1

0sinte

t1dt=+1X n=11n 2+ 1? f(t) = cht????t2[;] Z 0 cht:cos(nt)dt= (1)nshn 2+ 1? ????n2N? ?? ???? u n=Z +1

0t btct(t+n)dt

??? ??????? ? ??????? A >0 Z A

0t btct(t+n)dt=1n

Zn

0t btct

dtZ A+n

At btct

dt v n=nun? v nvn112n? ?? ??????? ?? ????? ??????? un? Z +1

0sintt

dt? ?? ???? ???? ???? ????x f(x) =Z +1 xsintt dt? 0 f(t)dt? I=Z +1 0sin 3tt 2dt? ????x >0? ?? ????

I(x) =Z

+1 xsin 3tt 2dt? ?? ???? ????sin3a= 3sina4sin3a? ??????? ???

I(x) =34

Z 3x xsintt 2dt? ?? ??????? ?? ? ??????? I? I=Z +1 0 ln1 +t2t 2 dt?

0dt3 + cos

x7!Z 2x xdtp1 +t2+t4? u n=r(1) +r(2) ++r(n)n 2 v n=(nr(1)) + (nr(2)) ++ (nr(n))n 2? ???? ????x2[1;+1[? ?? ????

F(x) =Z

x 1tpt 31dt?
C

1???]1;+1[?

??????? ?? ?????? ?? F??+1? yy 0=py 31?
????x >0? ?? ????

F(x) =+1X

n=0(1)nn+x? ??? ??????? F??? ?? ??????C1? ?? ??????C1?

F(x) +F(x+ 1)?

??? ??????? ? ???x >0

F(x) =Z

1 0t x11 +tdt? ?????? ?? ????? ??????? F?? ? ?? ??+1? f(x) =+1X n=11n 2+x2? ??? ??????? f??? ?????? ?? ?? ??????C1???R? ??+1? +1X n=11n 2=26 ??+1X n=11n 4=490 uv=vu? ??????? ???u??v??? ?? ??????? ?????? ?? ??????? v= (vn)?????? ??? v

0=u0??8n2N;vn=un+un12

trM= 0??M34M2+ 4M=On?

F:L(E)! L(E)?????? ???

F:f7!12

(fp+pf)? A

3+In=On?

??????? ??? ?? ????? ??A??? ?? ??????? A=0 B

B@3 652

16 52

110 83

03 2 01

C

CA??B=0

B

B@1 2 6 21

0 2 2 5

0 0 3 2

0 0 0 51

C CA? ?? ?????? ?? ???? ??(xInB)1??(xInC)1? P A? ?? ??????? ???x????? ??? ?????? ?????? ??A? ??????? tr(xInA)1=P0A(x)P A(x)? M=A A (0)A ????A2 Mn(R)?

8P2R[X]; P(M) =P(A)AP0(A)

(0)P(A) (f) =pfs I=ZZ D (1 +xy)dxdy ??D??????? ?? ?????? ????? ?? ??????O?? ?? ????? ?? ff=f? x2[0;1]f(x) =x

Keru= Imu???Imu= Keru??

Ker(u+u) = Keru\Keru?

f(x) =nX k=1(ekjx)ek

8x2En f0Eg;(f(x)jx)>0?

g 2=f1? f(x) =+1X n=0x n2? ????2]1;1[? P n(x) =nY k=0 1kx ???? ??? ?????? ??? ???? ??????P(x)? (E):8x2R;f(x) = (1x)f(x)? f(x) =f(0)P(x)? ????p;q2N? ?? ????

I(p;q) =Z

1 0 tp(1t)qdt? ????n0? ?? ???? a n=Z =4 0 tanntdt? ?? ? ???f(x)?? ????? ?? ?? ????? ??????? +1X n=0a nxn? f(x) =Z +1 0 et2cos(xt)dt? I n=Z 1

011 +tndt?

I n`? Z1 0 ln(1 +tn)dt=+1X k=1(1)k1k(nk+ 1)? Z 1 0 ln(1 +tn)dt n=1n Z n 0 f(t)dt? ????n;m2N? ?? ???? I n(m) =Z 1 0 xn(lnx)mdx? ?? ??????? Z1 0 xxdx=+1X n=1n n?

F(x) =Z

+1 0e te2tt cos(xt)dt? ????n1? ?? ???? I n=Z +1

0dt(1 +t3)n?

??????? ??? ? ??????? ?? ????n1? I n+1=3n13nIn? u n= ln(nIn)? X n11n In ????A2 Mn(R)? ??????? rg( tAA) = rgA? ??????? ??? ???? ????X2 Mn;1(R)? t

XAX2R+?

X=X? t X X

8x;y2E;(f(x)jy) =(xjf(y))?

AX=X? tXAX? ??????? 2iR? y

00+q(x)y= 0?

????? ??? ?????? ???? ??+1? w=f0gfg0? r

2= cos(2)?

f(x;y) =( sin(xy)jxj+jyj??(x;y)6= (0;0)

0??????

??? ???? ???? ?????? ???????n?? ????Ln=n!(2n)!(X21)n(n)? ??????? ? 8Q2Rn1[X];R1

1Ln(t)Q(t)dt= 0?

??? ?? ????un=135(2n1)246(2n)? ???? vn= (n+ 1)u2n? Q2n k=2 11k detA0? n2N?u n+1u n=n+an+b? lnun+1lnun ??????? ??? ?? ?????(un)??? ?? ?????? ?????? ????? ?2R??(vn)?? ????? ?? ????? ??????? v n=nun? lnvn+1lnvn? ?? ??????? ????? ??????A >0??? ??? u nAn u n=Z +1

0dt(1 +t2)n?

??E? ??? ?? ????Sn=Pn k=11pk ????n2N

1pn+12pn+ 1pn

1pn (e1;e2;e3)???A=0 @211 1 21 11 21 A f???? ????? ????? 0P n k=0(1)kt2kdt=4 +R1

0(1)nt2n+21+t2dt?

?? ??????? ? ??????? ?? Sn=Pn k=0(1)k2k+1?????n!+1? M n(R)? ????n2N? ??? ??????? ? ??????? x2R?y(x) = y(0) +Rx

0f(t)etdt

e x? E?? M

0= sup

x2R f(x)??M2= sup x2R f00(x)? ???? ????h >0?f0(x)2M0h +hM22 M

1= sup

x2R f0(x)2pM 0M2? ??? ????A2 Mn(R)??? ??????? ?? ???? ?? A=XtY A

2= tr(A)A?

f(x) =x2 ???]0;]? +1X n=1sin8nn ??? ????A2 Mn(C)????? ??? ? ???? ?? ????? ?????? ?????? ??A? A n=On? det(A+In)? det(A+M)?

8M2GLn(C);AM=MA=)det(A+M) = detM??

k=0 n kcos(k)? ?????? ?? ????? ?????? ??????? ?? ????? ?? ????? ?? ?????yn=xnn? ?? ????~x=~v+~w?~y=~w+~u??~z=~u+~v? ??????? ??? ?? ?? ???????(~u;~v; ~w)??? ????? ????? ?? ???????(~x;~y;~z)????? ?????? ??? ???? ????n2N? ?? ????Sn=Pn k=11n+k??S0n=Pn k=1(1)k1k ??????? ??? ? ??????? p >1?Rp+1 pdtt 1p Rp p1dtt ?? ??????? ?? ???? ???? ?? ????? (Sn)? ??????? ??? S02n=Sn?? ?? ??????? ?? ?????? ??(S0n)? a

1=n+b1=n2

n ??f? E? ??????? ??? ?? ???????x0;f(x0);:::;fn1(x0)??? ??? ???? ??E? ?????? ?? ??????? ?? f???? ????? ????? ?? ? ???C=g2 L(E)gf=fg? ?? ????A2 Mn(R)? ??? ???? ????n2N? ???? u n=(2n)!(2 nn!)2? lnun+1lnun? nX k=0ln(uk+1)ln(uk)?????? ??????? ? ??????? ?? ?? ????? (un)? nu n!+1? ?? ??????? ?? ?????? ?? ?? ????? ?? ????? ???????un? ??????? ??? v2nun+vn2 ?? ??????? ??? vn!`? ??????? ??? ???? ????n2N?Kerf= Kerfn???fn=ff f????n ??? ???? ??Imfn????n2N?

0p1t2dt?

???????M()? ??? ??????a2Rn f0;1g? ?? ????h(x) =ax???? ???? ????x? ???? f2S? ??????? ???h1fh=f? ?? ??????? ?? ?????? ??f(0)? M=0 @1 0aquotesdbs_dbs9.pdfusesText_15

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