2016/2017 Feuille 5 : Notations ∑ . 1) Calculer les sommes suivantes

BCP ST 1B
P
S1=
5
X
k=1
(k−3) , S2=
56
X
i=1
2, S3=
25
X
i=8
(1) , S4=
12
X
n=0
cos nπ
12 
(a+b)3a b
n
X
k=1 (k+ 1)3−k3
n
X
k=1
k2=n(n+ 1)(2n+ 1)
6
n n ∈N
n
X
k=1
k2=n(n+ 1)(2n+ 1)
6
n∈N
n
X
k=0
k3= n
X
k=0
k!2
n
X
k=0
k3=
n
X
k=0
(n−k)3
n
X
k=0
k
n
X
k=0
k2
n
X
k=1
(−1)k,
n2
X
k=n+1
(2)k,
n+1
X
k=2
(2k+ 1) ,
n
X
k=2
3k+2
22k+1
n(a, b)∈C2
an+1 −bn+1 = (a−b)
n
X
k=0
an−kbk
n
n
X
k=0
kn
k2
=n
2
n
X
k=0 n
k2
n
X
k=0 n
kn
X
k=0 n
k(−1)k
n
X
k=0
kn
kn
X
k=0 k
p
n
2n
X
k=1
(−1)k+1
k=
n
X
k=1
1
n+k

n
un=
n
X
k=0 n−k
k
u0u1u2u3u4u5u6
un+2 un+1 un
unn
(an)
n−1
X
k=1
(ak−ak+1)
n−1
X
k=1
(ak−2ak+1 +ak+2)
n
n−1
X
k=1
k(ak−ak+1) = n
X
k=1
ak!−nan
(an) (bn)
n
Sn=
n
X
k=0
akbkBn=
n
X
k=0
bk
n
X
k=0
akbk=anBn−
n−1
X
k=0
Bk(ak+1 −ak)