CAIE FP1 2008 June — Question 2

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2008
SessionJune
TopicSequences and series, recurrence and convergence

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. \(x = 1\).