Write down the expansion of \(\ln ( 1 + 2 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
State the range of values of \(x\) for which this expansion is valid.
Given that \(y = \ln \cos x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
Find the value of \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) when \(x = 0\).
Hence, by using Maclaurin's theorem, show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \cos x\) are
$$- \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 12 }$$
Find
$$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln ( 1 + 2 x ) } { x ^ { 2 } - \ln \cos x } \right]$$