I'm a math student on a quest of relearning Calculus with an emphasis of developing my intuition. My question is: why do some functions converge on the whole $\mathbb{R}$, and some don't?
Let me expand. So far, I found good intuitive approach about main characteristics of functions. Example: what does it mean if a function isn't uniformly continuous on some open interval $(a,b)$? Well, there could be two ''problems'' with that function: it's oscillating too wildly or it's growing too fast. What does it mean if a function isn't differentiable at some point $a$? Well, the function isn't really ''smooth'' at that point, it has a ''horn''. I hope you see what kind of intuition I hope to get.
Let's get to the point: Why doesn't $\ln(1+x)$ converge on whole $\mathbb{R}$? I could try to explain it like this. Every form of Taylor remainder (Cauchy, integral, Lagrange) looks something like this n-th derivative at some point $\cdot$ distance from the main point to the n-th power $\cdot \frac{1}{n!}$
Every time you derivate $\ln(1+x)$, a constant factor pops out, so it creates counterbalance to that $\frac{1}{n!}$, and the deciding factor is distance from the main point.
The problem is I don't understand what quality of a function ''there is a constant factor that grows when you take repeated derivatives'' represents? There is a similar story about $\arctan$ (also a function that has a finite radius of convergence). The similarity between $\arctan$ and $\ln$ is that both of functions grow very slowly after initial ''boom'', but I can't connect that with high-order derivatives?
Dual to that, $e^x$, $\cos$ and $\sin$ have a good characteristic: their constant factors are bounded, but I also can't connect that with some visual quality of those functions.
Does someone have a good picture about this? I hope I made my question clear.