I asked this question a while ago and I figured out my initial problem. I basically learned that convergence of series doesn't mean that it converges to that function since there could be error
My question now is, if there is no error between a function and Taylor series, then does that mean that Taylor series is equal to that function (at least on that interval?)
And, for example, for $e^x$ or $\sin(x)$. if I prove the error is zero, then do I need to prove the radius of convergence of that series (perhaps by ratio test?)?