If I want to check whether a Taylor series converges to its function f(x). Is it enough to check if the radius of convergence of the Taylor series is infinite?
Or do I have to use the remainder theorem and show the remainder converges to zero?
Edit: I'm still confused about convergence of taylor series. If I have a function f(x) which is defined on $x\in ]-R,R[$, and I found it's taylor series T(x) with a radius of convergence R. Doesn't the taylor series converge to f(x)?
Why do I necessarily need to check that the remainder $R_n(x)$ converges to zero for it to be true?
In other words, what does the taylor series converge to, if it doesn't converge to f(x)? (Because I thought if a taylor series was convergent it would always be towards f(x)? But that's probably where the whole issue is?).