2018 ACJC P1 Q2

Two series \({{S}_{n}}\) and \({{T}_{n-1}}\) are given by

\({{S}_{n}}=\sum\limits_{r=1}^{n}{\left( \frac{r}{{{2}^{r-1}}} \right)}\) and \({{T}_{n-1}}=\sum\limits_{r=1}^{n-1}{\left( \frac{r}{{{2}^{r}}} \right)}\).

By listing the terms of \({{S}_{n}}\) and \({{T}_{n-1}}\), write \({{S}_{n}}-{{T}_{n-1}}\) as a 3 geometric series and find \({{S}_{n}}-{{T}_{n-1}}\), leaving your answer in terms of \(n\).[3]

Hence find the value of \({{S}_{n}}-{{T}_{n-1}}\) as \(n\to \infty \). [1]