What's the meaning of being expressible as a convergent power series in a neighborhood of each point?

Question

The following pictures are from Lee's "Introduction to Smooth Manifolds".

What's the meaning of being expressible as a convergent power series in a neighborhood of each point? However, I only know convergent power series in real and complex fields.

Answer 1

I hope you have understood enough of the definition so far to understand that the geometry of the manifold is determined by overlapping patches each of which looks like a piece of real $n$-space. So where the patches overlap you have a transition map $T$ that is a function from $\mathbb{R}^n$ to itself. So you can ask how smooth $T$ is. Perhaps it has derivatives up to order $k$. In the best case, it is infinitely differentiable, and, moreover, it has a Taylor series expansion (in $n$ variables) that converges to the function. That's like $T(x) = 1/(1-x)$ in one variable: the Taylor series for that function converges to the function in a neighborhood of each point in the interval $(=1,1)$.