2012 SRJC P1 Q1

The graph of \(y=\frac{2}{1+x}\) for \(x\ge 0\), is shown in the diagram below. Region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the line \(x=1\) and the curve \(y=\frac{2}{1+x}\). The area of region \(R\) may be approximated by the total area, \(A\), of \(n\) rectangles, each of width \(\frac{1}{n}\), as shown in the diagram.

1. Show that \(A=\sum\limits_{r=0}^{n-1}{\frac{2}{n+r}}\).[2]
2. By considering the exact area of region \(R\), show that \(\sum\limits_{r=0}^{n-1}{\frac{1}{n+r}}>\ln 2\).[2]