What is an example such that $f(x)\neq \sum_{m=0}^{k-1} \frac{f^{(m)}(\alpha)}{m!} (x-\alpha)^m + \frac{f^{(k)}(\psi)}{k!} (x-\alpha)^k$?

Question

Let $f:[a,b]\rightarrow \mathbb{C}$ be a $C^{k-1}$ and assume $f^{(k-1)}$ is differentiable on $(a,b)$.

If the range of $f$ is real, then the usual taylor's theorem holds, but I'm not sure whether the theorem holds for complex function too.

The only thing I could prove is:

$|f(x) - \sum_{m=0}^{k-1} \frac{f^{(m)}(\alpha)}{m!} (x-\alpha)^m| ≦ |\frac{f^{(k)}(\psi)}{k!} (x-\alpha)^k|$.

Is there an example such that

$f(x)\neq \sum_{m=0}^{k-1} \frac{f^{(m)}(\alpha)}{m!} (x-\alpha)^m + \frac{f^{(k)}(\psi)}{k!} (x-\alpha)^k$?

Or, generally the equality holds?

Answer 1

I am not sure if you are asking about existence of Taylor series for infinitely differentiable function. Here is an example of real function that have all derivatives bounded but Taylor series is identically zero around 0: \begin{equation} f(x) = e^{-\frac{1}{x^2}}\end{equation}