Your question actually describes an infinite dimension division algebra - the field of Formal Laurent series. In particular, the elements of this field are just expressions of the form:
where the series can go on forever to the right, but may not have infinitely many terms of negative exponent (although can have arbitrarily many). Note that we're not concerning ourselves with convergence or anything like that - these are purely expressions that you manipulate by adding them coefficient wise and multiplying them by taking every pair of terms from the two series, taking their product, then collecting like terms (which is, for each coefficient, a finite process due to the fact that there are only finitely many negative terms included in any series).
It's sort of a pain to write out the exact formula for the multiplicative inverse of an element, but you can do it fairly nicely in two steps: First, note that every non-zero element is of the form
$$c\cdot x^n\cdot F$$
where $c$ is an element of the field we're taking our coefficients from and where $F$ is of the form $F=1+a_1x+a_2x^2+\ldots$. Since we can clearly invert $c$ as it's just a real number (or something like that) and we can invert $x^n$, all we need to do is invert $F$. We can do that by solving
which gives the equations, for each $n\geq 1$ that
which rearranges to say
after we pull out one term from the sum. We can then inductively figure out the power series inverse to any of the form $1+a_1x+a_2x^2+\ldots$ and extend that as you wish.
It's also worth noting that this construction gives a division ring whenever we take our coefficients from a division ring - so if we want something non-commutative, we could apply this construction to have quaternion coefficients.