Maclaurin Series (Edexcel A Level Further Maths: Core Pure): Exam Questions

$\mathrm{f}\left(x\right)=\frac{1}{2}\ln \left(\frac{2+x}{1-x}\right),-2<x<1$

Show that

$\mathrm{f}\left(x\right)=artanh\left(\frac{2x+1}{3}\right)$

Hence, or otherwise, show that

$\mathrm{f}\text{'}\left(x\right)=\frac{3}{2(2+x)(1-x)}$

Hence show that the Maclaurin series for $\mathrm{f}\left(x\right)$, up to and including the term in $x^2$, is

$\ln p+qx+r{x}^2$

where $p$, $q$ and $r$ are constants to be determined.