Integration Q3: Rectangle Under Curve

The diagram shows a sketch of the graph of \(y=\frac{1}{\sqrt{x}}.\)

By considering the shaded rectangle and the area of the region between the graph and the \(x\)-axis for \(n-1\le x\le n\) where \(n\ge 1,\) show that \(\frac{1}{\sqrt{n}}<2\left[ \sqrt{n}-\sqrt{\left( n-1 \right)} \right].\)

Deduce that \(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}<2\sqrt{n}.\)

Show also that \(\frac{1}{\sqrt{n}}>2\left[ \sqrt{\left( n+1 \right)}-\sqrt{n} \right]\) and hence, deduce that

\(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}>2\sqrt{\left( n+1 \right)}-2.\)