I am trying to decide when a function can be written as a Taylor series.
I think it exists if the following condition is met:
For a Taylor series of $f(x)$ about the point $a$
In the region $R$ containing both $x$ and $a$, the function $f(x)$ is single-valued with an infinite number of continuous derivatives that all exist.
Is this both a necessary and sufficient condition? If not then what is?
Best Answer
Any function which is infinitely differentiable at a point $a$ has a Taylor series at that point. Whether or not the Taylor series converges at any point other than $a$ is a different issue. But for the existence of a Taylor series all you need is the coefficients to exist, and these only require knowing the derivatives of the function at that point, so this is your sufficient condition. It is of course also necessary since if the function has a Taylor series, then the coefficients contain all higher derivatives at the point.