Factorielle et binôme de Newton Cours

n∈N∗
n!=1×2×3× · · · × (n−1) ×n n
0 ! = 1 n! (n+ 1)! = n!×(n+ 1)
n
p
n p
k∈ {0,1, . . . , n}k
nn
kk n
n
k=n! k! (n−k)! =n(n−1) . . . (n−k+ 1) k!.
n
kk
n k
n
n∈Nx, y ∈R
(x+y)n=n
0xn+n 1xn−1y+· · · +n n−1xyn−1+n nyn=
n
X
k=0 n
kxkyn−k
=n
0yn+n 1xyn−1+· · · +n n−1xn−1y+n nxn=
n
X
k=0 n
kxn−kyk.
n
k
k, n ∈Nk6nn
n−k=n k
n
0=n n= 1 n 1=n n−1=nn 2=n n−2=n(n−1) 2
k, n ∈Nk6n−1n
k+n k+ 1=n+ 1 k+ 1
n
kk n
aa
a
k
n0 1 2 3 4 5 6 7 8
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
7 1 7 21 35 35 21 7 1
8 18285670562881
p, q ∈Np6q up, up+1, . . . , uq−1, uq
q
Y
i=p
ui=up×up+1 × · · · × uq−1×uq.
n! =
n
Y
i=1
iePn
i=1 ui=
n
Y
i=1
euiu1, . . . , un>0 ln n
Y
i=1
ui!=
n
X
i=1
ln ui
n≥2 cosn(x) sinn(x)
cos(kx) sin(kx)k∈N∗
sin3(x)=eix−e−ix
2i 3
sin3(x) = 1
−8i(eix)3+ 3(eix)2(−e−ix) + 3(eix)(−e−ix)2+ (−e−ix)3
=−1
8ie3ix−3eix+ 3e−ix−e−3ix=−1 8i2i sin(3x)−3×2i sin(x)
=−1
4sin(3x) + 3 4sin(x).
n≥2 cos(nx) sin(nx)
cosk(x) sinl(x)k, l ∈N
cos(3x) = <eei(3x)sin(3x) = =mei(3x)
e3ix=eix3= (cos x+ i sin x)
3= cos3x+ 3 cos2x(i sin x) + 3 cos x(i sin x)2+ (i sin x)3
=cos3x−3 cos xsin2x+ i 3 cos2xsin x−sin3x.
cos(3x) = cos3x−3 cos xsin2x= cos3x−3 cos x(1 −cos2x)
= 4 cos3x−3 cos x,
sin(3x) = 3 cos2xsin x−sin3x= 3(1 −sin2x) sin x−sin3x
= 3 sin x−4 sin3x.

n!n∈ {0,1,2,...,7}
50 !
46 !
(2n+ 3)!
(2n+ 1)!
(n+ 1)!
(n−2)! +n! (n−1)!
(n−1)!
n!−n! (n+ 1)!
(2n)!
n!n∈Nn∈ {1,2,3,4}
n∈N∗
n
Y
k=1
(2k)=2nn!
n
Y
k=0
(2k+ 1) = (2n+ 1)!
2nn!
n>10 n!>362880 ×10n−9
n∈N∗
n
X
k=1
1
k!6
n
X
k=1
1
2k−1<2
n
n
k
n
5
2 50 2 50 49
(a+b)6(2x−1)5
PRP(x) = x4+ 2x3−1P(x+ 1)
a4b2c3a4b3c3(a−b+2c)9
cos6x x 7→ cos6x
cos(5x)P(cos x)P
1.0110 ≈1.105
0.99810−3
f:x7→ (1 + x)n
S1=
n
X
k=0 n
kS2=
n
X
k=0
(−1)kn
kS3=
n
X
k=0
kn
kS4=
n
X
k=0
1
k+ 1n k
u0= 1 n∈N∗un=−n un−1un
n
(x+ 1)2n= (x+ 1)n(x+ 1)n
n
X
k=0 n
k2
=2n
n
k
p=k+ 1 p+ 1−k p+ 1q
X
k=pk
p
p, q ∈Np6q
n∈N
n
X
k=1 k
1n>1
n
X
k=2 k
2n>2
n
X
k=0
k
n
X
k=0
k(k−1)
n
X
k=0
k2
X x1, . . . , xn
X pk=P{X=xk}16k6n
E(X) =
n
X
k=1
pkxk,V(X) = E(X−E(X))2=
n
X
k=1
pk[xk−E(X)]2.
V(X) = EX2−E(X)2EX2=
n
X
k=1
pkx2
k.
n∈N∗X
{1,2, . . . , n}X{1,2, . . . , n}
∀k∈ {1,2, . . . , n},P{X=k}=1
n.
1nE(X)V(X)
n∈N∗p∈]0,1[ X
n p X
{0,1, . . . , n}
∀k∈ {0,1, . . . , n},P{X=k}=n
kpk(1 −p)n−k.
n
E(X)V(X)