Expii

# Taylor Series of f(x) at x=a - Expii

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!).