I apologize as I am still somewhat unfamiliar with infinite series and knowing when and how we are allowed to use them, and especially with working with operators in this way. I imagine that I will be incorrect somewhere in this post or in its title.
I am working with finite-difference schemes for derivatives. I was reading a thesis on the subject when I came across the following material from the appendix of the book Numerical Analysis by Zdenek Kopal.
Suppose $\phi$ is a function of a single variable, $x$, which is discretized into a mesh $x_i = x_0 + ih$ defined by a spacing parameter $h$. We will use the notation $\phi(x + ih) = \phi(x_i) = \phi_i$. Note that this will be the case even for cases when $i$ is not an integer and therefore $x_i$ does not lie on the mesh.
Note that, by Taylor expansion,$$\begin{equation}\phi_{i+1} = \sum_{n=0}^{\infty} \frac{h^n}{n!}\phi^{(n)}_i = \sum_{n=0}^{\infty} \frac{(h\frac{\text{d}}{\text{d}x})^n}{n!}\phi_i = e^{h\frac{\text{d}}{\text{d}x}}\phi_i.\end{equation}\label{1}\tag{1}$$
Define an operator $\delta$ such that $\delta\phi_i = \phi_{i + \frac12} - \phi_{i - \frac12}$. Per the above Taylor expansion, we have$$\begin{equation}\delta\phi_i = (e^{\frac{h}{2}\frac{\text{d}}{\text{d}x}} - e^{-\frac{h}{2}\frac{\text{d}}{\text{d}x}})\phi_i,\end{equation}\label{2}\tag{2}$$so$$\begin{equation}\frac{\delta}{2} = \sinh\left(\frac h2 \frac{\text{d}}{\text{d}x}\right).\end{equation}\label{3}\tag{3}$$
Therefore we can write$$\begin{equation}\frac{\text{d}}{\text{d}x} = \frac{2}{h}\text{arcsinh}\left(\frac{\delta}{2}\right)\end{equation}\label{4}\tag{4}$$and especially (using the series expansion of arcsinh)$$\begin{equation}\frac{\text{d}}{\text{d}x} = \frac2h\left[\frac\delta2 - \left(\frac12\right)\frac{\delta^3}{2^3\cdot3} + \left({1 \cdot 3 \over 2 \cdot 4}\right)\frac{\delta^5}{2^5\cdot5}- \left({1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6}\right)\frac{\delta^7}{2^7\cdot7} + \dots\right].\end{equation}\label{5}\tag{5}$$
I am comfortable with \eqref{1} with the caveat that I don't understand why we are allowed to use all the infinite number of terms in the first step and expect equality. I am fully comfortable with both \eqref{2} and \eqref{3} as, in my mind, functions with operators as inputs are just shorthand for the full infinite series.
However, I see no reason why we should be able to deduce \eqref{4} and \eqref{5}. Why should we be able to transition from infinite series to function to inverse function back to infinite series and know that our result is correct? Another point of confusion for me is that since operators are not numbers, how would we even describe convergence of an infinite series of them? The radius of convergence of the series expansion of arcsinh (when defined with numbers) is finite, but is that of concern? How can I better understand what's going on here?
Later statements in the thesis involve a series expansion of $\left(1 + \frac{\delta^2}{4}\right)^{-\frac{1}2}$ and division by $\delta$ and another operator which I will not define here. I find these things to be just as confusing as what I have described above.
I realize I ask a lot of questions, but any amount of information regarding this, pointers to resources, or even the name of a topic I can research to learn about this would be immensely helpful to me.
