OCR FP1 2014 June — Question 6

Exam BoardOCR
ModuleFP1 (Further Pure Mathematics 1)
Year2014
SessionJune
TopicSequences and series, recurrence and convergence

6
  1. Show that \(\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 2 ) ^ { 2 } } \equiv \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\).
  2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 1 } ^ { n } \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\).
  3. Find \(\sum _ { r = 5 } ^ { \infty } \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\), giving your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.