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)).

4 questions

CAIE Further Paper 2 2024 June Q2
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 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.
WJEC Further Unit 4 2023 June Q5
  1. (a) Write down and simplify the Maclaurin series for \(\sin 2 x\) as far as the term in \(x ^ { 5 }\).
    (b) Using your answer to part (a), determine the Maclaurin series for \(\cos ^ { 2 } x\) as far as the term in \(x ^ { 4 }\).
  2. (a) Show that \(\tan \theta\) may be expressed as \(\frac { 2 t } { 1 - t ^ { 2 } }\), where \(t = \tan \left( \frac { \theta } { 2 } \right)\).
The diagram below shows a sketch of the curve \(C\) with polar equation $$r = \cos \left( \frac { \theta } { 2 } \right) , \quad \text { where } - \pi < \theta \leqslant \pi .$$ \includegraphics[max width=\textwidth, alt={}, center]{1d01b3d3-8a45-4c64-9d2f-ced042d8fba3-4_680_887_1171_580}
(b) Show that the \(\theta\)-coordinate of the points at which the tangent to \(C\) is perpendicular to the initial line satisfies the equation $$\tan \theta = - \frac { 1 } { 2 } \tan \left( \frac { \theta } { 2 } \right) .$$ (c) Hence, find the polar coordinates of the points on \(C\) where the tangent is perpendicular to the initial line.
(d) Calculate the area of the region enclosed by the curve \(C\) and the initial line for \(0 \leqslant \theta \leqslant \pi\).
AQA Further AS Paper 1 2019 June Q10
3 marks
10
  1. Using the definition of \(\cosh x\) and the Maclaurin series expansion of \(\mathrm { e } ^ { x }\), find the first three non-zero terms in the Maclaurin series expansion of \(\cosh x\). 10
  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).
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-15_2488_1716_219_153}