OCR FP2 2011 January — Question 2

Exam BoardOCR
ModuleFP2 (Further Pure Mathematics 2)
Year2011
SessionJanuary
TopicTaylor series
TypeMaclaurin series for hyperbolic inverse functions

2 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } x\).
  1. Show that \(\mathrm { f } ^ { \prime \prime \prime } ( x ) = \frac { 2 \left( 1 + 3 x ^ { 2 } \right) } { \left( 1 - x ^ { 2 } \right) ^ { 3 } }\).
  2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\), up to and including the term in \(x ^ { 3 }\).