Series for composite expressions

Questions that require finding series for functions of series, such as (ln(1+x))^2 or ln(e^(2x) + e^(-2x)), where the outer function must be applied to an inner series.

2 questions

OCR FP2 2009 January Q1
1
  1. Write down and simplify the first three terms of the Maclaurin series for \(\mathrm { e } ^ { 2 x }\).
  2. Hence show that the Maclaurin series for $$\ln \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)$$ begins \(\ln a + b x ^ { 2 }\), where \(a\) and \(b\) are constants to be found.
OCR Further Pure Core 1 2024 June Q9
9
  1. Find the Maclaurin series of \(( \ln ( 1 + x ) ) ^ { 2 }\) up to and including the term in \(x ^ { 4 }\). The diagram below shows parts of the graphs of the curves with equations \(y = ( \ln ( 1 + x ) ) ^ { 2 }\) and \(y = 2 x ^ { 3 }\). The curves intersect at the origin, \(O\), and at the point \(A\).
    \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-4_663_906_831_248} \section*{(b) In this question you must show detailed reasoning.} Use your answer to part (a) to determine an approximation for the value of the \(x\)-coordinate of \(A\). Give your answer to \(\mathbf { 2 }\) decimal places.