If $f$ and $g$ have the Taylor expansions given by
$$f(x) = f_{0} + f_1x + f_2x^2 + \mathcal{O}(x^3), $$
$$g(x) = g_0 + g_1x + g_2x^2 + \mathcal{O}(x^3), $$
is it possible to find the Taylor expansion of the product $f(x)g(x)$ up to $\mathcal{O}(x^3)$?
I was thinking of multiplying termwise, like this:
$$f(x)g(x) = f_0g_0 + f_1g_1x + f_2g_2x^2 + \mathcal{O}(x^3), $$
but I'm not so sure if that's allowed.
This is just a simple calculation, i.e. $$f(x)g(x)=f_0g_0 + (f_0g_1+f_1g_0)x+(f_0g_2+f_1g_1+f_2g_0)x^2+O(x^3)$$