Multiplication of two Taylor expansions I'm trying to calculate a Taylor expansion which is :  $\cos(x). exp(x)$  in the neighborhood of 0 in order 3
this is the result I got : 
$$\cos(x). exp(x) = \left(1-\frac{x²}{2}+\epsilon(x)x^3\right) . \left(1+x+\frac{x²}{2}+\frac{x^3}{6}+\epsilon(x)x^3\right)$$
And now I need to multiply the two expressions.
I think that there is a method where we use a table to multiply to Taylor expansion, but I don't know hw to do it.

Why don't you use regular definition
$$\hat f(x)=f(x=0)+f'(x=0)x+\frac 12 f''(x=0)x^2+\frac 16 f'''(x=0)x^3+O(x^4)$$
where
$$f(x=0)=e^x \cos(x)=1$$
$$f'(x=0)=e^x \cos(x)-e^x \sin(x)=1$$
$$f''(x=0)=-2e^x \sin(x)=0$$
$$f'''(x=0)=-2e^x \cos(x)-2e^x \sin(x)=-2$$
and
$$\hat f(x)=1+x-\frac 26 x^3+O(x^4)$$
