Use series to approximate numerical value

A question is this type if and only if it asks to substitute a specific value into a series to approximate a number (not an integral).

7 questions · Challenging +1.2

4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n
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OCR MEI FP3 2007 June Q3
24 marks Challenging +1.8
3 The curve \(C\) has equation \(y = \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } \ln x\), and \(a\) is a constant with \(a \geqslant 1\).
  1. Show that the length of the arc of \(C\) for which \(1 \leqslant x \leqslant a\) is \(\frac { 1 } { 2 } a ^ { 2 } + \frac { 1 } { 4 } \ln a - \frac { 1 } { 2 }\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(\boldsymbol { y }\)-axis.
  3. Show that the radius of curvature of \(C\) at the point where \(x = a\) is \(a \left( a + \frac { 1 } { 4 a } \right) ^ { 2 }\).
  4. Find the centre of curvature corresponding to the point \(\left( 1 , \frac { 1 } { 2 } \right)\) on \(C\). \(C\) is one member of the family of curves defined by \(y = p x ^ { 2 } - p ^ { 2 } \ln x\), where \(p\) is a parameter.
  5. Find the envelope of this family of curves.
OCR Further Pure Core 2 2021 November Q10
6 marks Challenging +1.2
10 In this question you must show detailed reasoning.
  1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(\mathrm { e } ^ { x } > 1 + x\).
  2. Hence, by using a suitable substitution, deduce that \(\mathrm { e } ^ { t } > \mathrm { e } t\) for \(t > 1\).
  3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(\mathrm { e } ^ { \pi }\) or \(\pi ^ { \mathrm { e } }\).
Pre-U Pre-U 9795/1 2016 June Q6
16 marks Challenging +1.2
6 The equation \(\sinh x + \sin x = 3 x\) has one positive root \(\alpha\).
  1. Show that \(2.5 < \alpha < 3\).
  2. By using the first two non-zero terms in the Maclaurin series for \(\sinh x + \sin x\), show that \(\alpha \approx \sqrt [ 4 ] { 60 }\).
  3. By taking the third non-zero term in this series, find a second approximation to \(\alpha\), giving your answer correct to 4 decimal places.
OCR MEI FP2 2009 June Q1
16 marks Standard +0.3
    1. Use the Maclaurin series for \(\ln(1 + x)\) and \(\ln(1 - x)\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln\left(\frac{1 + x}{1 - x}\right)\). State the range of validity of this series. [4]
    2. Find the value of \(x\) for which \(\frac{1 + x}{1 - x} = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places. [4]
  2. A curve has polar equation \(r = \frac{a}{1 + \sin \theta}\) for \(0 \leq \theta \leq \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
    1. By plotting suitable points, or otherwise, sketch the curve. [3]
    2. Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve. [5]
OCR MEI Further Extra Pure 2021 November Q6
8 marks Challenging +1.8
You are given that \(q \in \mathbb{Z}\) with \(q \geqslant 1\) and that $$S = \frac{1}{(q+1)} + \frac{1}{(q+1)(q+2)} + \frac{1}{(q+1)(q+2)(q+3)} + \cdots$$
  1. By considering a suitable geometric series show that \(S < \frac{1}{q}\). [3]
  2. Deduce that \(S \notin \mathbb{Z}\). [2]
You are also given that \(\mathrm{e} = \sum_{r=0}^{\infty} \frac{1}{r!}\).
  1. Assume that \(\mathrm{e} = \frac{p}{q}\), where \(p\) and \(q\) are positive integers. By writing the infinite series for \(\mathrm{e}\) in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that \(\mathrm{e}\) is irrational. [3]
SPS SPS FM Pure 2021 May Q9
6 marks Standard +0.8
  1. Using the Maclaurin series for \(\ln(1 + x)\), find the first four terms in the series expansion for \(\ln(1 + 3x^2)\). [2]
  2. Find the range of \(x\) for which the expansion is valid. [1]
  3. Find the exact value of the series $$\frac{3^1}{2 \times 2^2} - \frac{3^2}{3 \times 2^4} + \frac{3^3}{4 \times 2^6} - \frac{3^4}{5 \times 2^8} + \ldots$$ [3]
SPS SPS FM Pure 2022 February Q12
6 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(e^x > 1 + x\). [2]
  2. Hence, by using a suitable substitution, deduce that \(e^t > et\) for \(t > 1\). [1]
  3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(e^{\pi}\) or \(\pi^e\). [3]