5. (a) Use a Maclaurin series to find a quadratic approximation for \(\ln ( 1 + 2 x )\).
(b) Find the percentage error in using the approximation in part (a) to calculate \(\ln ( 1.2 )\).
(c) Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid.
[0pt] [BLANK PAGE] \section*{6. In this question you must show detailed reasoning.} In this question you may assume the results for $$\sum _ { r = 1 } ^ { n } r ^ { 3 } , \quad \sum _ { r = 1 } ^ { n } r ^ { 2 } \quad \text { and } \quad \sum _ { r = 1 } ^ { n } r$$ Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n ^ { 2 } \left( 2 n ^ { 2 } - 1 \right)$$ [BLANK PAGE]