Infinite Geometric Series Formula Derivation

If by derive, you mean go from the summation to the fraction representation, you probably identified the best ways of doing it. However, here's one non-rigorous way to get the result going the other way, i.e., starting with the fraction

$$ \frac{a_0}{1-r} $$

Some have observed that you can write the Taylor series for that at $r=0$. Another way is to use synthetic division or polynomial long division. It's hard to typeset here, but I'll give you the flavor as best I can. Think of the long division algorithm we learned in grade school, where you are generating the terms on the top one at a time as you are dividing the dividend by the term $1-r$, multiplying the newly generated term by the divisor, subtracting, and iterating:

$$ \begin{matrix} {} & a_0 & +a_0 r & +a_0 r^2 & +a_0 r^3 &+\cdots\\\hline 1-r)&a_0\\ {}&-a_0&+a_0 r\\\hline {}&{}&a_0 r\\ {}&{}&-a_0 r & +a_0 r^2\\\hline {}&{}&{}&a_0 r^2\\ {}&{}&{}&-a_0 r^2&+a_0 r^3\\\hline {}&{}&{}&{}&a_0 r^3\\ {}&{}&{}&{}&-a_0 r^3&+a_0 r^4\\\hline {}&{}&{}&\vdots&{} \end{matrix} $$

The process never terminates, but does successively give additional terms of the expansion you are asking about. After conjecturing the series generated represents the function, you of course have to check convergence and prove the formula's correctness, but it works out in this case.