Suppose the function f has derivatives of all orders on an interval centered at the point a. The Taylor series for f centered at ais
A Taylor series centered at 0 is called a MacLaurin Series

Convergence of Taylor Series

Let f have derivatives of all orders on an open interval I containing a. The Taylor series for f centered at a converges to f, for all x in I,if and only if for all x in I, where
is the remainder at x (with c between x and a)