I have a function defined only by it's taylor series: $$f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!}x^k$$
Obviously, integer derivatives can be defined as $$\frac{d^n}{dx^n} f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{(k-n)!}x^{k-n}$$
However, this completely fails if $n$ is not an integer and $x=0$; every term vanishes, which is obviously incorrect. Is there a better way to take a fractional derivative of a Taylor series?