$ \lim_{K\rightarrow\infty}\frac{(1-\epsilon)^K}{1+(1-\epsilon)^K}\frac{\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l} K \\ i \end{array}\right)\left[\left(\frac{2\epsilon-\epsilon^2}{(1-\epsilon)^2}\right)^i-\left(\frac{\epsilon}{1-\epsilon}\right)^i\right]}{(1-\epsilon)^K\left[1+\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l} K \\ i \end{array}\right)\left(\frac{2\epsilon-\epsilon^2}{(1-\epsilon)^2}\right)^i\right]+\left[1+\sum_{i=1}^{\frac{K-1}{2}}\left(\begin{array}{l} K \\ i \end{array}\right)\left(\frac{\epsilon}{1-\epsilon}\right)^i\right]}=? $ where $1>\epsilon>0$. I wonder especially when $\epsilon\rightarrow 0$ but $\epsilon\neq 0$
I even dont know if this is a difficult or an easy question for a mathematician. If you could comment on this matter, I will be happy.