Use Maclaurin's series expansion for \(\ln ( 1 + x )\) to show that the first three terms of the Maclaurin's series expansion of \(\ln ( 1 + 3 x )\) are
$$3 x - \frac { 9 } { 2 } x ^ { 2 } + 9 x ^ { 3 }$$
15
Julia attempts to use the series expansion found in part (a) to find an approximation for \(\ln 4\)
Julia's incorrect working is shown below.
$$\begin{array} { r }
\text { Let } 1 + 3 x = 4
3 x = 3
x = 1
\end{array}$$
$$\text { So } \begin{aligned}
\ln 4 & \approx 3 \times 1 - \frac { 9 } { 2 } \times 1 ^ { 2 } + 9 \times 1 ^ { 3 }
& \approx 3 - 4.5 + 9
& \approx 7.5
\end{aligned}$$
Explain the error in Julia's working.
15
Use \(x = - \frac { 1 } { 6 }\) in the series expansion found in part (a) to find an approximation for \(\ln 4\) Fully justify your answer.