2018 ACJC P1 Q7 [Modified]

Express \(\frac{1}{x\left( x+1 \right)}\) in partial fractions.[1]

The graph of \(y=\frac{1}{x\left( x+1 \right)}\), for \(0\le x\le n\), is shown in the diagram. Rectangles, each of width \(1\) unit, are drawn below the curve from \(x=1\) to \(x=n\), where \(n\ge 3\).

By considering \(\sum\limits_{x=a}^{b}{\frac{1}{x\left( x+1 \right)}}\) where \(a\) and \(b\) are constants to be found, find the total area of the \(n-1\) rectangles in terms of \(n\).[3]

Hence, show that \(\frac{1}{2}-\ln 2<\frac{1}{n+1}+\ln \left( 1-\frac{1}{n+1} \right)\) for all \(n\ge 3\).[3]