Why the general formula of Taylor series for $\ln(x)$ does not work for $n=0$? I need to find the taylor series for $\ln(x)$ about $a = 2$, and I have find the following solution, but I don't understand why the general formula does not work for $n = 0$.


The general formula would be $\frac{(-1)^{0+1} (-1)!}{2^0} = -(-1)!$. The factorial of $-1$ is undefined, so the general formula isn't defined for $n=0$. However $f^{(0)}(2) = f(2) =\ln 2$. That's what is meant by "not working".
Note that this is closely related to $\int x^{n-1}\ \mathrm dx = \frac1n x^n$ wich is also not defined for $n=0$ but $\int x^{-1} \ \mathrm dx = \ln x$.
