Question

: f(x)=1/(2-x), centrado en x=0
Ejercicio 1: determine los polinomios de Taylor desde el grado cero (0) hasta el grado cinco (5).

Answer 1

Los polinomios de Taylos desde el grado cero hasta el cinco es:

p(x) = 1/2 + (1/4)x + (1/8)x² + (1/16)x³ + (1/16)x⁴ + (1/64)x⁵

función

f(x) = 1/(2-x)

x= 0

p(x) = f(0) + f'(0)*(x-0) + f''(0)*(x-0)²/2! + f'''(0)*(x-0)³/3! + f⁴(0)*(x-0)⁴/4! + f⁵(0)*(x-0)⁵/5!

Hallar las derivadas

f'(x) = $\frac{1}{\left(2-x\right)^2}$

f''(x) = $\frac{2}{\left(2-x\right)^3}$

f'''(x)= $\frac{6}{\left(2-x\right)^4}$

f⁴(x) = $\frac{24}{\left(2-x\right)^5}$

f⁵(x) =$\frac{120}{\left(2-x\right)^6}$

Evaluando en x=0

f(0) = 1/2

f'(0) = 1/4

f''(0) = 2/8 = 1/4

f'''(0)= 6/16= 3/8

f⁴(0) = 24/16=3/2

f⁵(0) = 120/64= 15/8

p(x) = 1/2 + 1/4*x + 1/4*x²/2! + 3/8*x³/3! + 3/2*x⁴/4! + 15/8*x⁵/5!

p(x) = 1/2 + (1/4)x + (1/8)x² + (1/16)x³ + (1/16)x⁴ + (1/64)x⁵