3 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).
- Using the method of differences, or otherwise, show that \(S _ { n } = \ln \frac { n + 2 } { 2 ( n + 1 ) }\).
Let \(S = \sum _ { r = 1 } ^ { \infty } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\). - Find the least value of \(n\) such that \(\mathrm { S } _ { \mathrm { n } } - \mathrm { S } < 0.01\).