Como calcular series de Taylor

Hallamos la serie de Taylor en un valor cercano a 2, digamos en 0

$$\begin{align}&f(0)=0\\&f'(0)=1\\&f''(0)=0\\&f'''(0)=-1\\&\\&f^{iv}(0)=0\\&\\&\sin(x)=x-\dfrac{1}{3!}x^3+\dfrac{1}{5!}x^5-\dfrac{1}{7!}x^7+\dfrac{1}{9!}x^9+E_{11}\\&\\&\sin(2)\approx2-\dfrac{2^3}{3!}+\dfrac{2^5}{5!}-\dfrac{2^7}{7!}+\dfrac{2^9}{9!}\\&\\&\sin(2)\approx\dfrac{2578}{2835}\\&\\&\sin(2)\approx 0.9093\\&\end{align}$$

$$\begin{align}&T(x)=f(c)(x-c)+f'(c)(x-c)+\dfrac{f''(c)}{2!}(x-c)^2+\dfrac{f'''(x-c)}{3!}(x-c)^3+\dfrac{f^{iv}(x-c)}{4!}(x-c)^4\\&\\&f(x)=(1+x^2)^{1/2}\\&f'(x)=x(1+x^2)^{-1/2}\\&f''(x)=(1+x^2)^{-3/2}\\&f'''(x)=-3x(1+x^2)^{-5/2}\\&f^{iv}(x)=3(4x^2-1)(1+x^2)^{-7/2}\end{align}$$

etcétera.