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Given: $$f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ i \ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i$$ f_2(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K\\ i
\end{array} \right) \left(-1+\frac{1}{a}\right)^i prove or disprove that $$f_3(a)=\frac{f_1(a)}{f_2(a)}$$ is an increasing funtion of $a$, where $-1\ -1< r < 0$ and $0.5 < a < 1$
Given: $$f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\ i \ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i$$ f_2(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K\\ i
\end{array} \right) \left(-1+\frac{1}{a}\right)^i prove or disprove that $$f_3(a)=\frac{f_1(a)}{f_2(a)}$$ is an increasing funtion of $a$, where \$-1\