Combining or manipulating standard series

Questions that require adding, subtracting, or using algebraic identities to combine multiple standard series or derive one series from another (e.g., finding cos²x from cos 2x, finding sin²x from cos 2x, or adding e^(1+x²) + e^(1-x)).

5 questions · Standard +0.5

4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
Sort by: Default | Easiest first | Hardest first
CAIE Further Paper 2 2024 June Q2
4 marks Moderate -0.3
2 Find the Maclaurin's series for \(\mathrm { e } ^ { 1 + x ^ { 2 } } + \mathrm { e } ^ { 1 - x }\) up to and including the term in \(x ^ { 2 }\).
OCR MEI Further Pure Core 2019 June Q5
5 marks Standard +0.3
5 Using the Maclaurin series for \(\cos 2 x\), show that, for small values of \(x\), \(\sin ^ { 2 } x \approx a x ^ { 2 } + b x ^ { 4 } + c x ^ { 6 }\),
where the values of \(a , b\) and \(c\) are to be given in exact form.
Pre-U Pre-U 9795/1 2013 June Q2
4 marks Challenging +1.2
2 Use the standard Maclaurin series expansions given in the List of Formulae MF20 to show that $$\frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \equiv \tanh ^ { - 1 } x \text { for } - 1 < x < 1$$
AQA Further AS Paper 1 2019 June Q10
6 marks Standard +0.3
  1. Using the definition of \(\cosh x\) and the Maclaurin series expansion of \(e^x\), find the first three non-zero terms in the Maclaurin series expansion of \(\cosh x\). [3 marks]
  2. Hence find a trigonometric function for which the first three terms of its Maclaurin series are the same as the first three terms of the Maclaurin series for \(\cosh(ix)\). [3 marks]
WJEC Further Unit 4 2023 June Q5
7 marks Standard +0.8
  1. Write down and simplify the Maclaurin series for \(\sin 2x\) as far as the term in \(x^5\). [2]
  2. Using your answer to part (a), determine the Maclaurin series for \(\cos^2 x\) as far as the term in \(x^4\). [5]