5.5 1) 1·2 + 2·3 + 3·4 + 4·5 +. . .+n(n+ 1) =
n
X
k=1
k(k+ 1)
2)
n
X
k=1
k(k+1) =
n
X
k=1
k2+k=
n
X
k=1
k2+
n
X
k=1
k = n(n+ 1) (2n+ 1)
6 +n(n+ 1) 2
= n(n+ 1) (2n+ 1) + 3n(n+ 1)
6 = n(n+ 1) (2n+ 1) + 3 6
= n(n+ 1)
2 (n+2)
z }| { (2n+ 4)
6 = 2n(n+ 1) (n+ 2)
6 = n(n+ 1) (n+ 2) 3
Analyse : séries Corrigé 5.5