Method of differences with logarithmic terms

3 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).

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

2 Given that $$u _ { n } = \ln \left( \frac { 1 + x ^ { n + 1 } } { 1 + x ^ { n } } \right)$$ where \(x > - 1\), find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(N\) and \(x\). Find the sum to infinity of the series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ when

1. \(- 1 < x < 1\),
2. \(x = 1\).

1. (i) Find the value of

$$\sum _ { r = 4 } ^ { \infty } 20 \times \left( \frac { 1 } { 2 } \right) ^ { r }$$ (3)
(ii) Show that $$\sum _ { n = 1 } ^ { 48 } \log _ { 5 } \left( \frac { n + 2 } { n + 1 } \right) = 2$$

1. (a) Use the method of differences to prove that for \(n > 2\)

$$\sum _ { r = 2 } ^ { n } \ln \left( \frac { r + 1 } { r - 1 } \right) \equiv \ln \left( \frac { n ( n + 1 ) } { 2 } \right)$$ (4)
(b) Hence find the exact value of $$\sum _ { r = 51 } ^ { 100 } \ln \left( \frac { r + 1 } { r - 1 } \right) ^ { 35 }$$ Give your answer in the form \(a \ln \left( \frac { b } { c } \right)\) where \(a , b\) and \(c\) are integers to
be determined.

9

1. Show that, for \(r > 0\), $$\ln ( r + 2 ) - \ln r = \ln \left( 1 + \frac { 2 } { r } \right)$$ 9
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \ln \left( 1 + \frac { 2 } { r } \right) = \ln \left( \frac { 1 } { 2 } ( n + a ) ( n + b ) \right)$$ where \(a\) and \(b\) are integers to be found.