Use series to find error or validity

A question is this type if and only if it asks about the error in a series approximation, percentage error, or range of validity of a series.

5 questions · Standard +0.5

4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
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OCR MEI Paper 3 2023 June Q15
2 marks Challenging +1.2
15 The expression given in line 34 is used to calculate \(\sum _ { r = 1 } ^ { 6 } \frac { 1 } { r }\).
Show that the error in the result is less than \(1.5 \%\) of the true value.
OCR MEI Further Pure Core 2021 November Q5
6 marks Standard +0.8
5
  1. Use a Maclaurin series to find a quadratic approximation for \(\ln ( 1 + 2 x )\).
  2. Find the percentage error in using the approximation in part (a) to calculate \(\ln ( 1.2 )\).
  3. Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid.
AQA Further AS Paper 1 2024 June Q15
7 marks Standard +0.3
15
  1. Use Maclaurin's series expansion for \(\ln ( 1 + x )\) to show that the first three terms of the Maclaurin's series expansion of \(\ln ( 1 + 3 x )\) are $$3 x - \frac { 9 } { 2 } x ^ { 2 } + 9 x ^ { 3 }$$ 15
  2. Julia attempts to use the series expansion found in part (a) to find an approximation for \(\ln 4\) Julia's incorrect working is shown below. $$\begin{array} { r } \text { Let } 1 + 3 x = 4 \\ 3 x = 3 \\ x = 1 \end{array}$$ $$\text { So } \begin{aligned} \ln 4 & \approx 3 \times 1 - \frac { 9 } { 2 } \times 1 ^ { 2 } + 9 \times 1 ^ { 3 } \\ & \approx 3 - 4.5 + 9 \\ & \approx 7.5 \end{aligned}$$ Explain the error in Julia's working.
    15
  3. Use \(x = - \frac { 1 } { 6 }\) in the series expansion found in part (a) to find an approximation for \(\ln 4\) Fully justify your answer.
OCR MEI Further Pure Core Specimen Q7
11 marks Standard +0.3
  1. Use the Maclaurin series for \(\ln(1 + x)\) up to the term in \(x^3\) to obtain an approximation to \(\ln 1.5\). [2]
    1. Find the error in the approximation in part (i). [1]
    2. Explain why the Maclaurin series in part (i), with \(x = 2\), should not be used to find an approximation to \(\ln 3\). [1]
  3. Find a cubic approximation to \(\ln\left(\frac{1+x}{1-x}\right)\). [2]
    1. Use the approximation in part (iii) to find approximations to • \(\ln 1.5\) and • \(\ln 3\). [3]
    2. Comment on your answers to part (iv) (A). [2]
SPS SPS FM Pure 2023 November Q5
6 marks Moderate -0.3
  1. Use a Maclaurin series to find a quadratic approximation for \(\ln(1 + 2x)\). [1]
  2. Find the percentage error in using the approximation in part (a) to calculate \(\ln(1.2)\). [3]
  3. Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid. [2]