2020 VJC P1 Q3

It is given that \(\mathrm{f}(r)=\ln \left( 2+{{\alpha }^{r+1}} \right)\), where \(\alpha \) is a constant and \(-1<\alpha <1\).

By considering \(\mathrm{f}(r+1)-\mathrm{f}(r)\), find \(\sum\limits_{r=1}^{n}{\ln \left( \frac{2+{{\alpha }^{r+2}}}{2+{{\alpha }^{r+1}}} \right)}\) in terms of \(n\) and \(\alpha \).[3]

Hence, give a reason why the series

\(\ln \left( \frac{2+{{\alpha }^{4}}}{2+{{\alpha }^{3}}} \right)+\ln \left( \frac{2+{{\alpha }^{5}}}{2+{{\alpha }^{4}}} \right)+\ln \left( \frac{2+{{\alpha }^{6}}}{2+{{\alpha }^{5}}} \right)+...\)

converges. Given that \(\alpha =0.7\), find the value of the sum to infinity.[4]