Polinomi de Taylor

Per trobar el polinomi de Taylor hem de conèixer el valor de la derivada primera, segona i tercera de $f (x)$ al punt $x_0=0$.

Calculem:

$$\left\{ \begin{array}{l} f(x)=\dfrac{1}{1+x}=(1+x)^{-1} \\ f'(x)=-1(1+x)^{-2} \\ f''(x)=2(1+x)^{-3} \\ f'''(x)=-6(1+x)^{-4} \end{array} \right. \Rightarrow \left\{ \begin{array}{l} f(0)=1 \\ f'(0)=-1 \\ f''(0)=2 \\ f'''(0)=-6 \end{array} \right. $$

Per tant, el polinomi de Taylor és: $$\begin{array}{rl} T_3(x)=&f(x_0)+\dfrac{f'(x_0)}{1!}(x-x_0)+ \dfrac{f''(x_0)}{2!}(x-x_0)^2+ \dfrac{f'''(x_0)}{3!}(x-x_0)^3 \\ =& 1+\dfrac{-1}{1}(x-0)+\dfrac{2}{2}(x-0)^2+\dfrac{-6}{6}(x-0)^3 \\ =& 1-x+x^2-x^3 \end{array} $$