It seems to me, that many of the critical ideas in Lie Theory has a feeling of Taylor series behind it. For example, we have the concept of linearizing a continous group to get a lie algebra, and then getting back the group by exponentiation. The way the linearlizing seems quite similar to how we linearly approximate functions:
$$ f(a+h) \approx \left[ (1+h \frac{d}{dx}) f(x) \right]_{x=a}$$
And, how we get back the total taylor series seem somewhat similar to how we have to exponentiate the group element (here we do $h \frac{d}{dx}$. Coincidence, or something deeper?
What I got so far:
$\mathbb{R^d}$ is in general a lie group, we even in higher dimension, we have this similar ideas to the said above applicable.
