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SERIES DE TAYLOR E DE MACLAURIN´
Notas de aula - Pedro Miranda^1
Defini¸c˜ao
Pn(x) = F (a) + F ′(a) 1! .(x − a) + F ′′(a) 2! .(x − a)^2 +... F (n)(a) n! .(x − a)n
Ou seja,
P (x) = c 0 + c 1 (x − a) + c 2 (x − a)^2 + c 3 (x − a)^3 +...
|x − a| < R
cn = F (n)(a) n!
Exemplo 1. F (x) =
∑^ ∞
n=
F (n)(a) n! .(x − a)n
= F (a) + F ′(a) 1! .(x − a) + F ′′(a) 2! .(x − a)^2 + F ′′′(a) 3! .(x − a)^3...
P/a = 0
F (x) =
∑^ ∞
n=
F (n)(0) n! .xn^ = F (0) +
F ′(0)
.x +
F ′′(0)
.x^2 +...
Obs.: A express˜ao acima ´e conhecida como s´erie de Maclaurin
Exemplo 2. Ache a s´erie e o raio de convergˆencia da fun¸c˜ao: F (x) = ex.
F (x) = ex^ ⇒ F (n)^ = ex, F (0)(0) = e^0 = 1 P/∀n
∑^ ∞
n=
F (n)(0) n! .xn^ =
∑^ ∞
n=
xn n!
x 1!
x^2 2!
x^3 3!
Convergˆencia: An = xn n!
, ent˜ao:
An+ An
xn+ (n + 1)!
n! xn^
|x| n + 1
Converge P/∀x ⇒ Rc = ∞.
ex^ =
∑^ ∞
n=
xn n!
F (x) = lim n 7 →∞ Tn(x) ⇒ Rn = F (x) − T(x)
F (x) = Tn(x) + Rn(x), Rn = resto.
(^1) e-mail: [email protected]
nlim 7 →∞ Tn(x) = lim n 7 →∞[F^ (x)^ −^ Rn(x)] =^ F^ (x)^ −^ nlim 7 →∞ Rn(x)
Teorema 1. Se f (x) = Tn(x) + Rn(x), onde Tn ´e o polinˆomio de Taylor.
Ent˜ao: (^) nlim→∞ Rn(x) = 0, p/|x − a| < R, (intervalo).
Desigualdade de Taylor
Se |F (n+1)(x)| ≤ n, p/|x − a| ≤ D, ent˜ao:
Rn(x) satisfaz: |Rn(x)| ≤ n (n + 1)! .|x − a|n+1, p/|x − a| ≤ D.
|F ′′(x)| ≤ n ⇒ F ′′(x) ≤ n, p/a ≤ x ≤ a + D, temos:
∫ (^) x
a
F ′′(t)dt ≤
∫ (^) x
a
ndt ⇒ F ′(x) − F ′(a) ≤ n(x − a) ou F ′(x) ≤ F ′(a) + n(x − a).
∫ (^) x
a
F ′(t)dt ≤
∫ (^) x
a
[F ′(a) + n(t − a)]dt
F (x) − F (a) ≤ F ′(a).(x − a) + n. (x − a)^2 2
mas R 1 (x) = F (x) − T 1 (x) = F (x) − F (a) − F ′(a).(x − a)
R 1 (x) ≤ n 2 .(x − a)^2
Exemplo 3. sin x = x − x^3 3!
x^5 5!
x^7 7! +... (−1)n. x^2 n−^1 (2n − 1)!
∑^ ∞
n=
(−1)n. x^2 n+ (2n + 1)!
Usando Maclaurin
Exemplo 4. F (0) +
F ′(0)
.x +
F ′′(0)
.x^2 +
F ′′′(0)
+... = x − x^3 3!
x^5 5!
x^7 7!
∑^ ∞
n=
(−1)n. x^2 n+ (2n + 1)!
A desigualdade
|Rn(x)| ≤ n (n + 1)! .|xn+1| = |x|n+ (n + 1)!
Exemplo 5. Ache a raiz quadrada de 4,8. √ 4 , 8 ⇒
x, a = 4.
Temos:
F ′(x) =
x ; F ′′(x) = −
4 x
x ; F ′′′(x) =
8 x^2
x
