Limites de sucesiones: polinomios.

Departamento de Matemáticas
Sucesiones tipo Polinomio.
(
Caso general: lim a ⋅ n p + b ⋅ n p−1 + c ⋅ n p−2 + ...
n →∞
con a , b, c,... ∈ R
)
y p∈N
a ⋅ n p + b ⋅ n p−1 + c ⋅ n p −2 + ... =
⎛ a ⋅ n p b ⋅ n p−1 c ⋅ n p −2
⎞
a ⋅ n p ⋅ ⎜⎜
+
+
+ ... ⎟⎟ =
p
p
p
a⋅n
a ⋅n
⎝ a⋅n
⎠
b
c
⎛
⎞
a ⋅ n p ⋅ ⎜1 +
+
+ ... ⎟ ⇒
2
⎝ a ⋅n a ⋅n
⎠
(
)
b
c
⎛
⎞
lim a ⋅ n p + b ⋅ n p−1 + c ⋅ n p− 2 + ... = lim a ⋅ n p ⋅ ⎜1 +
+
+ ... ⎟ =
2
n →∞
n →∞
⎝ a ⋅n a ⋅n
⎠
a ⋅ ∞ p ⋅ (1 + 0 + 0 + ...) = a ⋅ ∞ = ±∞
El ± depende del signo de a.
A efectos prácticos se puede decir que el límite de la sucesión
a n = a ⋅ n p + b ⋅ n p −1 + c ⋅ n p −2 + ... coincide con el límite de la sucesión a n = a ⋅ n p
(
)
(
lim a ⋅ n p + b ⋅ n p−1 + c ⋅ n p−2 + ... = lim a ⋅ n p
n →∞
n →∞
Ejemplos
• lim 3n 2 − 50n + 34 = lim 3n 2 = +∞
(
)
( )
lim ( −5n + 14n − n + 1) = lim ( −5n ) = −∞
lim ( − n + 2300n − n + 5n + 3n + 1) = lim ( − n ) = −∞
n →∞
•
•
n →∞
3
2
3
n →∞
5
n →∞
4
3
n →∞
2
5
n →∞
)