As we all (should) know, the Maclaurin series is a special case of the Taylor series when the Taylor series is centered around 0. This is the canonical definition of the Maclaurin series:
$$ f(x) = \sum_{n=0}^{\infty}\dfrac{f^{(n)}(0)}{n!}x^n $$
My question is this. Should it be that:
$$ \dfrac{f(x)}{g(x)} = {\dfrac{\sum^{\infty}_{n=0}{\dfrac{f^{(n)}(0)}{n!}}x^n}{\sum^{\infty}_{n=0}{\dfrac{g^{(n)}(0)}{n!}}x^n}} $$
or that:
$$ \sum^\infty_{n=0}{\dfrac{\dfrac{d^{(n)}}{d^{(n)}x}{\dfrac{f(x)}{g(x)}}}{n!}}x^n $$
Intuitively speaking, I find the former equation much more appealing, not only because it seems logical, but also because the latter equation would be much much harder to solve for quotients.
The same question applies to products.
