Direct substitution into standard series

Questions that ask to write down or find a Maclaurin series by directly substituting an expression (like x², 2x, or x³) into a single standard series (e^x, ln(1+x), sin x, cos x, etc.).

5 questions · Standard +0.0

4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
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OCR FP2 2007 June Q2
5 marks Moderate -0.5
2
  1. Given that \(\mathrm { f } ( x ) = \sin \left( 2 x + \frac { 1 } { 4 } \pi \right)\), show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \sqrt { 2 } ( \sin 2 x + \cos 2 x )\).
  2. Hence find the first four terms of the Maclaurin series for \(\mathrm { f } ( x )\). [You may use appropriate results given in the List of Formulae.]
OCR FP2 2014 June Q2
5 marks Moderate -0.3
2 It is given that \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\).
  1. Using the standard Maclaurin expansion for \(\ln ( 1 + x )\), write down the first four terms in the expansion of \(\mathrm { f } ( x )\), stating the set of values of \(x\) for which the expansion is valid.
  2. Hence find the exact value of $$1 - \frac { 1 } { 2 } \left( \frac { 1 } { 2 } \right) ^ { 2 } + \frac { 1 } { 3 } \left( \frac { 1 } { 2 } \right) ^ { 4 } - \frac { 1 } { 4 } \left( \frac { 1 } { 2 } \right) ^ { 6 } + \ldots .$$
OCR MEI Further Pure Core 2024 June Q10
6 marks Moderate -0.3
10
  1. Write down the first three terms of the Maclaurin series for \(\ln \left( 1 + x ^ { 3 } \right)\).
  2. Use these three terms to show that \(\ln ( 1.125 ) \approx \frac { n } { 1536 }\), where \(n\) is an integer to be determined.
  3. Charlie uses the same first three terms of the series to approximate \(\ln 9\) and gets an answer of 147, correct to 3 significant figures. However, \(\ln 9 = 2.20\) correct to 3 significant figures. Explain Charlie's error.
OCR Further Pure Core 1 2018 September Q10
6 marks Standard +0.3
10
  1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
  2. Find the range of \(x\) for which the expansion is valid.
  3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots .$$
WJEC Further Unit 4 2024 June Q2
13 marks Standard +0.8
The function \(f\) is defined by \(f(x) = \cosh\left(\frac{x}{2}\right)\).
  1. State the Maclaurin series expansion for \(\cosh\left(\frac{x}{2}\right)\) up to and including the term in \(x^4\). [2]
Another function \(g\) is defined by \(g(x) = x^2 - 2\). The diagram below shows parts of the graphs of \(y = f(x)\) and \(y = g(x)\). \includegraphics{figure_2}
  1. The two graphs intersect at the point A, as shown in the diagram. Use your answer from part (a) to find an approximation for the \(x\)-coordinate of A, giving your answer correct to two decimal places. [5]
  2. Using your answer to part (b), find an approximation for the area of the shaded region enclosed by the two graphs, the \(x\)-axis and the \(y\)-axis. [6]