5.11 1) Le terme général de la série s’écrit 1 k(k+ 1).
2) 1 k −
1
k+ 1 = (k+ 1)−k
k(k+ 1) = 1 k(k+ 1)
3) sn= Xn
k=1
1 k(k+ 1) =
Xn
k=1
1 k −
1 k+ 1
= 1 1−
1
| {z }2
k=1
+1 2 −
1
| {z }3
k=2
+1 3 −
1
| {z }4
k=3
+. . .+ 1 n −
1 n+ 1
| {z }
k=n
= 1− 1 n+ 1
4) S = lim
n→+∞
sn = lim
n→+∞
1− 1
n+ 1 = 1− lim
n→+∞
1
n+ 1 = 1−0 = 1
Analyse : séries Corrigé 5.11