I'm trying to find the fifth order polynomial for the two below questions. I've tried to manually compute it but got lost past the third order polynomial. Is there any general formula for 5th order Taylor polynomial ? Both these equations are centered around the origin
1)$$e^{-xy}\arctan(y)$$ 2)$$\frac{\cos(2xy)}{3-6x}$$
Expand $e^t=P_5(t)+O(t^6)$ and $\cos(t)=Q_5(t)+O(t^6)$ and $\dfrac{1}{3-6t}=\dfrac{1}{3} \sum\limits_{k=0}^\infty (2t)^k=R_5(t)+O(t^6)$. Also, $\arctan(t)=\int \dfrac{1}{1+t^2} dt=\int \sum\limits_{k=0}^{\infty}(-t^2)^k dt=\sum\limits_{k=0}^\infty \int (-1)^kt^{2k} dt=S_5(t) + O(t^6).$ The rest is algebra.