The Taylor series of a function \(f(x)\) at a point \(x=a\) is a power series centered at \(x=a\) modeled off of Taylor polynomial approximations. It has a radius of convergence like any other power series. In ideal situations, the Taylor series will not only converge, but converge to the original function on an open interval containing \(a\). In this case, the Taylor series is sometimes called the (power or Taylor) series expansion centered at \(x=a\). (Furthermore, the Taylor series is actually the unique power series converging to \(f(x)\) --- which makes Taylor series all the more natural!).