Riemann Sum of curve \(2x+{{2}^{x}}\)

The diagram shows that the curve with equation \(y=2x+{{2}^{x}}\) for \(0\le x\le 1\). The region bounded by the curve, the axes and the line \(x=1\) is denoted by \(S\). The area of region \(S\)can be approximated by the total area of \(n\) rectangles, \(A\), as shown in the diagram. Each rectangle is of width \(\frac{1}{n}\).

1. \(\frac{1}{n}\left\{ 2\left( \frac{1}{n}+\frac{2}{n}+...+\frac{n-1}{n} \right)+\left( 1+{{2}^{\frac{1}{n}}}+{{2}^{\frac{2}{n}}}+...+{{2}^{\frac{n-1}{n}}} \right) \right\}.\)[2]
2. Hence, find \(A\) in the form \(p+\frac{q}{n}+\frac{r}{n\left( {{2}^{\frac{1}{n}}}-1 \right)}\), where \(p\), \(q\) and \(r\) are constants to be determined.[3]
3. By considering region \(S\), find the limit of \(A\) as \(n\to \infty \), giving your answer to 2 decimal places.[2]