1666724786535_math analyse

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II. Calcul des sommes :
∑
+∞
κ=0 x κ {
+∞
n ′ existe pas si x ≤ −1
1
1−x
∑ k(k − 1)x k−2
κ=2
+∞
κ=1
si − 1 ≤ x ≤ 1
+∞ si x ≥ 1
=
2
(1 − x)
3
si − 1 < x < 1
∑ kx k 1
=
(1 − x) 2 si − 1 < x < 1
+∞
∑ xk
k=0
+∞
xk−1
+∞
xk−2
k! = ∑ (k − 1)! = ∑ = 2 x
(k − 2)!
k=1
Application:
k=2
1. ∑ +∞
κ=0 (−2) κ = n ′ existe pas
2. ∑+∞
κ=0 3 κ = +∞
3. ∑
+∞
( 1 2 )κ = 1
= 2
κ=0
1− 1 2
4. ∑
+∞
( 1 4 )κ = 1
κ=0
1− 1 4
= 4 3