Answer:
x/(1-x)
Explanation:
We start by induction. The first 2 terms are x/(1-x^2) and x^2/(1-x^4), respectively. When we add them together using special factorizations, we get: (x^3+x^2+x)/(1-x^4). We quickly realize that the numerator is a geometric series. Applying the following formula, we can simplify our expression to (x^4-x)/(1-x^4)(x-1). Now, we introduce our 3rd term, and add that to our sum. That gets us: (x^7+x^6+...+x)/(1-x^8). This is another Geometric series, and we apply the exact same formula, for this new fraction: (x^8-x)/(1-x^8)(x-1). We realize this is a pattern, and we can utilize this pattern for the sum of the first n terms.
We look at the exponents. For the first 3 terms, the Degree is 8, for the first 2 terms, the Degree is 4. We take note that this looks like x^2^n.
Using that our general formula IS: (attached to this text) (x^2^n-x)/(1-x^2^n)(x-1). Now, when 0<x<1 and n approaches infinity, what do we get?
all terms x^2^n are so small, it's basically redundant, and can be crossed off. This leaves us with -x/(x-1), which can be simplified to x/(1-x).