- The Taylor series expansion of \(f ( x )\) about \(x = a\) is given by
$$f ( x ) = f ( a ) + ( x - a ) f ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } f ^ { ( r ) } ( a ) + \ldots$$
Given that
$$y = ( 1 + \ln x ) ^ { 2 } \quad x > 0$$
- show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 2 \ln x } { x ^ { 2 } }\)
- Hence find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\)
- Determine the Taylor series expansion about \(x = 1\) of
$$( 1 + \ln x ) ^ { 2 }$$
in ascending powers of ( \(x - 1\) ), up to and including the term in \(( x - 1 ) ^ { 3 }\)
Give each coefficient in simplest form. - Use this series expansion to evaluate
$$\lim _ { x \rightarrow 1 } \frac { 2 x - 1 - ( 1 + \ln x ) ^ { 2 } } { ( x - 1 ) ^ { 3 } }$$
explaining your reasoning clearly.