It is given that $y=\sin [\ln (1+2 x)]$.
(i) Show that $(1+2 x)^2 \frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}+2(1+2 x) \frac{\mathrm{d} y}{\mathrm{~d} x}+k y=0$, where $k$ is a constant to be found.
Hence, find the first three non-zero terms of the Maclaurin expansion for $y$.


(ii) Using standard series from the List of Formulae (MF26), verify the correctness of your result from part (i) up to and including the term in $x^3$.


Explain why the expansion is not valid when $x=-\frac{1}{2}$.