Could you give me some hint or further advice me about the papers that can solve this solution
$\displaystyle{\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n-2}\log(p+(1-p)e^{ab(1-\frac{b(1-(bc)^{i})}{1-bc})-(bc)^ic})}$
assuming that bc < 1 and b $\in$ (0,1).
In my work, I find by using lower bound and upper bound of them, but I want to find the general form of this term if it could be ? Would you give me some advice or hint ? Thank you
This is $\lim\limits_{n\to\infty}\dfrac{1}{n}\sum\limits_{i=1}^{n-2}A_i=\lim\limits_{n\to\infty}A_n$ by Stolz–Cesàro. So, the answer must be $$\log\left(p+(1-p)e^{ab\big(1-b/(1-bc)\big)}\right).$$