Suites numériques (10)
3- Etude de quelques types de suites
3.1 Suite de la forme (np) avec p∈ℚ*
3.1.1 Propriété (1)
Si p∈IN
lim +∞ |
n² | = +∞ | lim +∞ |
n³ | = +∞ |
lim +∞ |
√(n) | = +∞ | lim +∞ |
np | = +∞ |
3.1.2 Propriété (2)
Si p∈ℤ-*
lim +∞ |
np | = 0 |
| Exemple | lim +∞ |
n-7 | = 0 |
3.1.4 Proprité (3)
Soit p∈ℚ*
| 1) Si p > 0 | 2) Si p < 0 | |||||
|---|---|---|---|---|---|---|
lim +∞ |
np | = +∞ | lim +∞ |
np | = 0 |
Exemples
lim +∞ |
n2/3 | = | lim +∞ |
³√(n)² = +∞ |
lim +∞ |
n-5/2 | = | lim +∞ |
√(n)-5= 0 |
Exercice 1 tp
Calculer les limlites suivantes
lim +∞ |
3n²+2n-5 | et | lim +∞ |
-2n³-2n²+7 |
Correction
lim +∞ |
3n² = +∞ et | lim +∞ |
2n-5 = +∞ |
| donc | lim +∞ |
3n²+2n-5 = +∞ |
lim +∞ |
-2n³ = -∞ et | lim +∞ |
-2n²+7 = -∞ |
| donc | lim +∞ |
-2n³-2n²+7 = - ∞ |
Exercice 2 tp
| Calculer | lim +∞ |
5n²-3n+4 |
Correction
+∞ - ∞ forme indéterminée
Soit n∈IN*
| 5n²-3n+4 = | 5n²(1- | 3 | + | 4 | ) |
| 5n | 5n² |
Puisque
lim +∞ | (1- | 3 | + | 4 | ) = 1-0+0 = 1 |
| 5n | 5n² |
alors
lim +∞ | 5n²-3n+4 = | lim +∞ | 5n² = + ∞ |
Exercice 3 tp
| Calculer | lim +∞ | (2√(n) +1)(1-5n) |
Correction
lim +∞ | (2√(n) +1) = +∞ |
lim +∞ | (1-5n) = -∞ |
| donc | lim +∞ | (2√(n) +1)(1-5n) = -∞ |
Exercice 4 tp
Calculer les limlites suivantes
lim +∞ |
n² | lim +∞ |
n²-3n+2 | |
| 2n²-5 | n-2 | |||
lim +∞ |
3n-2 | lim +∞ |
n²-1 | |
| n+1 | n4+1 | |||
lim +∞ |
n-√n | lim +∞ |
n-√(2n²+n) | |
| n |