Definition:Taylor Series

Definition

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:

$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({x}\right)$

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

See also

Taylor's Theorem

Source of Name

This entry was named for Brook Taylor.

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 15.4$