Definition:Taylor Series

Let $$f$$ be a real function which is smooth on the open interval $$\left({a \, . \, . \, b}\right)$$.

Let $$\xi \in \left({a \, . \, . \, b}\right)$$.

Then the Taylor series expansion about the point $$\xi$$ is $$\sum_{n=0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({x}\right)$$.

It is not necessarily the case that this power series is convergent with sum $$f \left({x}\right)$$.