This question and the comments and answers it received encouraged me to ask this question, although I know that there will be some people who think that this belongs in the math forum. But I think that this topic is more relevant to mathematical physicists than to pure mathematicians.
Motivation: One of the answers to this question explains that if $f\colon\mathbb C\to\mathbb C$ is suitable function and $A\colon\text{Dom}(A)\subset H\to H$ is a suitable operator, we can define \begin{equation} f(A):=\int_{\mathbb C}f\,\mathrm{d}P_A \end{equation} where $P_A\colon B(\mathbb C)\to B(H)$ is a measure. However, it's much easier to understand the definition in terms of convergent series, e.g. in the case of the exponential or the logarithm. (In statistical physics, $S=k_B\langle\ln\rho\rangle$ is the entropy, when $\rho$ is the density operator.) That's why I'd like to know:
Is it also possible to write $f(A)$ in terms of a converging series when $f$ has a taylor expansion around some point?
In case that the answer is yes, I also wonder if there is a relatively easy way to see how the integral and the series are equivalent. (As far as I know, integrals - even $\int_{\mathbb C}f\,\mathrm{d}P_A$ - can be expressed as limit of some series, so maybe that would be a good starting point).
Examples: The expression \begin{equation} \sum_{n=0}^\infty \frac{1}{n!} A^n \end{equation} makes sense whenever $A$ is an element of a complete normed space and converges to $\mathrm{e}^A=\int_{\mathbb C}\text{exp}\,\mathrm{d}P_A$ when $A$ is suitable operator (source).
It is even known that \begin{equation} \left(\sum_{k=1}^N (-1)^{k+1}\frac{(A-\text{id})^k}{k}\right)_{N\in\mathbb N} \end{equation} converges to $\text{ln}(A)$ under certain circumstances (see here and here), so I was wondering if there is a general rule. That is, if we have \begin{equation} f(x)=\sum_{n=0}^{\infty}a_n(x-b)^n, \end{equation} in a neighborhood of $b$, is \begin{equation} f(A)=\sum_{n=0}^{\infty}a_n(A-b\cdot\text{id})^n? \end{equation}