6
\includegraphics[max width=\textwidth, alt={}, center]{323ac7a5-4690-441d-87fc-325a393098fa-10_585_1349_258_358} The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + 2 \mathrm { x } } }\) for \(x > 0\), together with a set of \(( n - 1 )\) rectangles of unit
width. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + 2 r } } < \ln \left( n + 1 + \sqrt { n ^ { 2 } + 2 n } \right) + \frac { 1 } { 3 } \sqrt { 3 } - \ln ( 2 + \sqrt { 3 } )$$