I am currently working on "my own" system of infinitesimals. They are pretty much analogous to dual numbers, except for the fact that mine are not nilpotent. My main question concerns possible extensions of the exponential function, but before that, I will give some background information on how these numbers work.
I have a working notion of addition, multiplication and division. My numbers have tuple-representations, where a position in a tuple corresponds to the power of the number $\varepsilon$. Every tuple of length n is a subset of $\mathbb{R}^n$. Some examples:
$(1)=1\varepsilon^0=1$
$(1,0,2)=1+0\varepsilon^1+2\varepsilon^2=1+2\varepsilon^2$
Addition is done component-wise (if a tuple is too short, one can extend it with zeroes, i.e. $(1,0,1)=(1,0,1,0,0,0,0,0,0,0....)$ until it is of appropriate length), and the multiplication is just Cauchy-multiplication. "Rational" numbers work like you would expect, so expressions such as $\displaystyle \frac{\varepsilon^2-1}{\varepsilon}$ are defined. All in all, currently I have a field of rational functions in the variable $\varepsilon$ called $\mathbb{R}[\varepsilon]$. So far, this is just algebraic. My topological extension so far is the following:
$f(\varepsilon)\prec g(\varepsilon) \iff \exists a\in\mathbb{R}[f(x)<g(x)\ \forall x\in(0, a)]$
One can use this new ordered relation to prove intuitive results, such as $0\prec\varepsilon \prec u$ for all positive, real $u$.
Currently, I am trying to figure out how I could extend my numbers to the exponential function, i.e. have a function $\displaystyle\exp:\mathbb{R}[\varepsilon] \to \mathbb{R}(\varepsilon), x\mapsto \sum_{k=0}^{\infty}\frac{x^{k}}{k!}$, for some extended set $\mathbb{R}(\varepsilon)$ which also includes powers of $e$.
This seems like it would be a very useful function to have, as it would work nicely with the definitions already in place (power series are defined). But I have no idea how one could extend the function. For example, how could one be sure that it actually converges for infinitesimal arguments? I was thinking perhaps you could claim something such as:
"If a series converges for all $x\in(0, a)$ for some real $a$, then it will also converge for all infinitesimal $x$".
This seems logical as it would capture the idea that infinitesimals behave like appreciable, small numbers (same idea as for the extended ordered relation).
I am new to mathematics so I am sort of lost. Any suggestions would be appreciated!
