Reasons for Taylor Series Convergence

Question

I just started learning about convergence with Taylor series'. I am trying to clarify to myself what makes a Taylor series not converge? What property is it that sometimes allows it to converge everywhere and sometimes not? Any answers would be appreciated.

Answer 1

It's not quite clear to me what kind of properties you are looking for. But if you study complex analysis, you'll learn that the Taylor series converges out to the nearest singularity of the function in the complex plane. That explains, for example, why the Maclaurin series for $f(x)=1/(1+x^2)$ only has radius of convergence $R=1$ despite the function being nice for all real $x$: there are singularies at $x=\pm i$, at distance $1$ from the origin.