Maclaurin series for hyperbolic inverse functions

Questions asking to find Maclaurin series by differentiation for inverse hyperbolic functions such as tanh^(-1)(x).

1 questions · Standard +0.8

4.07e Inverse hyperbolic: definitions, domains, ranges4.08a Maclaurin series: find series for function
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OCR FP2 2011 January Q2
8 marks Standard +0.8
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 \(x ^ { 3 }\).