SEAMO 2027 - Paper E - SET 01 - Self-study Question 17

Let $f(x)=\frac{1+10x}{10-100x}$ and let $f^n=\underbrace{f\circ f\circ\cdots\circ f}_{n\text{ compositions}}$. Find $f(\frac12)+f^2(\frac12)+f^3(\frac12)+\cdots+f^{6000}(\frac12)$.