Composition of two standard series

Questions that require substituting one Maclaurin series into another standard series (e.g., e^(e^x - 1), ln(1 + sin x), cosh(ix)) involving nested functions.

3 questions · Standard +0.8

4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
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CAIE Further Paper 2 2023 November Q3
6 marks Challenging +1.2
3 Find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 } { 2 } e ^ { x } \right)\) in the form \(\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }\), giving the exact values of the constants \(a , b\) and \(c\).
OCR FP2 2015 June Q2
4 marks Standard +0.3
2 It is given that \(\mathrm { f } ( x ) = \ln ( 1 + \sin x )\). Using standard series, find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).
Edexcel CP2 2023 June Q2
6 marks Standard +0.8
  1. (a) Write down the Maclaurin series of \(\mathrm { e } ^ { x }\), in ascending power of \(x\), up to and including the term in \(x ^ { 3 }\) (b) Hence, without differentiating, determine the Maclaurin series of
$$\mathrm { e } ^ { \left( \mathrm { e } ^ { x } - 1 \right) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.