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Series · CAPE Pure Mathematics Unit 2

125 past-paper questions on Series, part of Sequences, Series and Approximations, from every CAPE Pure Mathematics Unit 2 paper on Quelpr.

Practise Series
  1. 3(a)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Use the fact that 1/r - 1/(r + 1) = 1 / (r(r + 1)) to show that S_n = sum_{r=1}^n 1/(r(r + 1)) = 1 - 1/(n + 1).
  2. 3(a)(ii)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Deduce that as n -> infinity, S_n -> 1.
  3. 3(b)10 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2The common ratio, r, of a geometric series is given by r = 5x / (4 + x^2). Find ALL the values of x for which the series converges.
  4. 6(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2Find the total number of visitors for the first n days.
  5. 6(b)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2The exhibition closed after 10 days. Determine how many people visited during the period for which it was opened.
  6. 6(b)(iv)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2005 · Paper 2If the exhibition had been kept opened indefinitely, what would be the maximum number of visitors?
  7. 3(a)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Show that the terms of sum_{r=1}^m ln 3^r are in arithmetic progression.
  8. 3(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Find the sum of the first 20 terms of this series.
  9. 3(a)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2006 · Paper 2Hence, show that sum_{r=1}^{2m} ln 3^r = (2m^2 + m) ln 3.
  10. 3(b)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Find the n-th term of S.
  11. 3(b)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Show that S is a geometric progression.
  12. 3(b)(iii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Find the first term and common ratio of S.
  13. 3(b)(iv)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Deduce the sum to infinity of S.
  14. 4(b)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Show that P = 570.
  15. 4(b)(ii)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2007 · Paper 2Find, in terms of n, 1 \le n \le 12, an expression for the remaining debt on the loan after John has paid the n-th instalment.
  16. 3(b)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2A GP with first term a and common ratio r has sum to infinity 81 and the sum of the first four terms is 65. Find the values of a and r.
  17. 3(c)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Write down the first FIVE terms in the power series expansion of \ln(1 + x), stating the range of values of x for which the series is valid.
  18. 3(c)(ii)a)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Using the result from (c)(i) above, obtain a similar expansion for \ln(1 - x).
  19. 3(c)(ii)b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2Hence, prove that \ln\left(\frac{1 + x}{1 - x}\right) = 2\left(x + \frac{1}{3}x^3 + \frac{1}{5}x^5 + \dots\right).
  20. 3(a)(i)a)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Show that T = S.
  21. 3(a)(i)b)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Deduce that S = \frac{1}{2} n(n + 1).
  22. 3(a)(ii)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Use the principle of mathematical induction to prove that \sum_{r=1}^n r^2 = \frac{1}{6} n(n + 1)(2n + 1).
  23. 3(a)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2Hence, prove that \sum_{r=1}^n 2r(3r + 1) = 2n(n + 1)^2.
  24. 3(b)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Find the range of values of x for which the common ratio r of a convergent geometric series is \frac{2x - 3}{x + 4}.
  25. 3(c)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Express f(r) - f(r + 1) in terms of r.
  26. 3(c)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Hence, or otherwise, find S_n = \sum_{r=1}^n \frac{4}{(r + 1)(r + 2)}.
  27. 3(c)(iii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Deduce the sum to infinity of the series in (c)(ii) above.
  28. 4(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Write down the first FOUR non-zero terms of the power series expansion of \ln(1 + 2x), stating the range of values of x for which the series is valid.
  29. 4(b)(ii)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Use Maclaurin's theorem to obtain the first THREE non-zero terms in the power series expansion in x of \sin 2x.
  30. 4(b)(iii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2009 · Paper 2Hence, or otherwise, obtain the first THREE non-zero terms in the power series expansion in x of \ln(1 + \sin 2x).
  31. 3(a)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2If S_n denotes the series formed by summing the first n terms of the sequence, find S_n in terms of n.
  32. 4(a)(ii)a)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Given that r is a positive integer and f(r) = \frac{1}{r!}, show that f(r) - f(r + 1) = \frac{r}{(r + 1)!}.
  33. 4(a)(ii)b)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Hence, or otherwise, find the sum S_n = \sum_{r=1}^{n} \frac{r}{(r + 1)!}.
  34. 4(a)(ii)c)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2Deduce the sum to infinity of S_n in (ii) b) above.
  35. 3(iv)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2The sum, S_n, of the first n terms of a series is given by S_n = n(3n - 4). Show that the series is an Arithmetic Progression (A.P.) with common difference 6.
  36. 4(a)(i)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Show that r + 1 + \frac{1}{r} = \frac{13}{3}.
  37. 4(a)(ii) a)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Hence, find the value of r.
  38. 4(a)(ii) b)1 mark· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find the value of a.
  39. 4(a)(ii) c)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Find the sum to infinity of the G.P.
  40. 4(b)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Expand \frac{2}{e^x + e^{-x}}, |x| < 1 in ascending powers of x as far as the term in x^4.
  41. 4(c)(i)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Express f(r) - f(r + 1) in terms of r.
  42. 4(c)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Hence, or otherwise, find S_n = \sum_{r=1}^n \frac{3}{r(r + 1)(r + 2)}.
  43. 4(c)(iii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2Deduce the sum to infinity of the series in (c)(ii).
  44. 3(a)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Hence, calculate n if S_n = 177\,146.
  45. 3(b)(ii)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Prove, by mathematical induction, that \sum_{r=1}^n u_r = \frac{1}{6}n(n + 1)(2n + 7), \forall n \in \mathbb{N}.
  46. 3(c)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Use Maclaurin's Theorem to find the first three non-zero terms in the power series expansion of \cos 2x.
  47. 3(c)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2Hence, or otherwise, obtain the first two non-zero terms in the power series expansion of \sin^2 x.
  48. 3(b)(i)a)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Let k > 0 and let f(k) = \frac{1}{k^2}. Show that f(k) - f(k+1) = \frac{2k+1}{k^2(k+1)^2}.
  49. 3(b)(i)b)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Show that \sum_{k=1}^n \left( \frac{1}{k^2} - \frac{1}{(k+1)^2} \right) = 1 - \frac{1}{(n+1)^2}.
  50. 3(b)(iii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Hence, or otherwise, prove that \sum_{k=1}^\infty \frac{2k+1}{k^2(k+1)^2} = 1.
  51. 3(c)(i)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Obtain the first four non-zero terms of the Taylor Series expansion of \cos x in ascending powers of (x - \frac{\pi}{4}).
  52. 3(c)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2Hence, calculate an approximation to \cos(-\frac{\pi}{16}).
  53. 3(a)(i)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Prove, by mathematical induction, that for n \in \mathbb{N}, S_n = 1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \dots + \frac{1}{2^{n-1}} = 2 - \frac{1}{2^{n-1}}.
  54. 3(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Hence, or otherwise, find \lim_{n \to \infty} S_n.
  55. 3(b)14 marks· CAPE Pure Mathematics Unit 2 · May/June 2014 · Paper 2Find the Maclaurin expansion for f(x) = (1 + x)^2 \sin x up to and including the term in x^3.
  56. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1For -1 < 2n < 1, \sum_{r=0}^{\infty} (2n)^r =
  57. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1Which of the following series are arithmetic series? \begin{align*} \text{I.} & \quad \sum_{r=1}^n (7 + 3r) \\ \text{II.} & \quad \sum_{r=1}^n 2(3^r) \\ \text{III.} & \quad \sum_{r=1}^n \log_{10}(r + 1) \\ \text{IV.} &…
  58. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The sum to infinity of a geometric series is \frac{1}{1 - 2x}. The range of x is
  59. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1Let a_n and S_n denote respectively, the value of the n^{\text{th}} term and the n^{\text{th}} partial sum of a series. The value of S_{n+2} - S_n when calculated on the series is
  60. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1Given that S_n = \sum_{i=1}^n \left(\frac{1}{i} - \frac{1}{i+1}\right), \lim_{n\to\infty} S_n is
  61. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1If \sum_{n=2}^\infty 2^{-n} = a, then a is
  62. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1The Maclaurin series for \sin x, up to the term in x^3, is
  63. 3(b)(i)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Express the nth partial sum S_n of the series in sigma notation.
  64. 3(b)(ii)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Hence, given that \sum_{n=1}^\infty \frac{1}{n^2} converges to \frac{\pi^2}{6}, show that S_n diverges as n \to \infty.
  65. 3(c)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Use the method of mathematical induction to prove that \sum_{r=1}^n r(r-1) = \frac{n(n^2-1)}{3}.
  66. 4(a)(i)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Obtain the Maclaurin series expansion for g(x) up to and including the term in x^4.
  67. 4(a)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2Hence, estimate g(0.2) correct to three decimal places.
  68. Q171 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1For -1 < 2n < 1, \sum_{r=0}^{\infty} (2n)^r =
  69. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1The sum to infinity of the geometric series 16 + 12 + 9 + \dots is
  70. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1\sum_{r=1}^n \left(\frac{1}{r} - \frac{1}{r + 1}\right) =
  71. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1Given that \sum_{k=1}^n k(k + 1) = S_n, then, for m < n, \sum_{k=m+1}^n k(k + 1) =
  72. 3(b)(i)7 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Show that S_n = (n(n^2 - 1)) / 3.
  73. 3(b)(ii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 2Hence, or otherwise, evaluate sum_{r=10}^20 r(r - 1).
  74. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Let a_n and S_n denote respectively, the value of the n^{\text{th}} term and the n^{\text{th}} partial sum of a series. The value of S_{n+2} - S_n when calculated on the series is
  75. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Which of the following series are arithmetic series? I. \sum_{r=1}^n (7 + 3r) II. \sum_{r=1}^n 2(3^r) III. \sum_{r=1}^n \log_{10}(r + 1) IV. \sum_{r=1}^n \log_{10} 3^{(r+1)}
  76. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The sum to infinity of a geometric series is \frac{1}{1 - 2x}. The range of x is
  77. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1The sum of the first n terms of a geometric series is \left[1 - \left(\frac{1}{2}\right)^n\right]. The value of the SECOND term is
  78. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Given that S_n = \sum_{i=1}^n \left(\frac{1}{i} - \frac{1}{i+1}\right), \lim_{n\to\infty} S_n is
  79. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 1Given that \sum_{k=1}^n k(k + 1) = S_n, then, for m < n, \sum_{k=m+1}^n k(k + 1) =
  80. 3(a)(i)6 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Determine the Taylor series expansion about x = 2 of the function f(x) = \ln(5 + x) up to and including the term in x^3.
  81. 3(a)(ii)2 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Hence, obtain an approximation for f(7) - \ln(7).
  82. 3(b)(i)9 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Use mathematical induction to prove that 1^3 + 2^3 + \ldots + n^3 = \frac{1}{4}n^2(n+1)^2, for n \in \mathbb{N}.
  83. 3(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Hence, or otherwise, show that \sum_{i=1}^{2n+1} i^3 = (2n + 1)^2(n + 1)^2.
  84. 3(b)(iii)5 marks· CAPE Pure Mathematics Unit 2 · May/June 2017 · Paper 2Use the results of Parts (b)(i) and (ii) to show that \sum_{i=1}^{n+1} (2i - 1)^3 = (n + 1)^2(2n^2 + 4n + 1).
  85. Q161 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1For -1 < 2n < 1, \sum_{r=0}^{\infty} (2n)^r =
  86. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The sum to infinity of a geometric series is \frac{1}{1 - 2x}. The range of x is
  87. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1Let a_n and S_n denote, respectively, the value of the n^{\text{th}} term and the n^{\text{th}} partial sum of a series. The value of S_{n+2} - S_n when calculated on the series is
  88. Q221 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1\sum_{r=1}^n \left(\frac{1}{r} - \frac{1}{r+1}\right) =
  89. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1Given that \sum_{k=1}^n k(k+1) = S_n, then, for m < n, \sum_{k=m+1}^n k(k+1) =
  90. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The coefficient of x^4 in the Taylor series expansion of f(x) = \cos x about x = 0 is
  91. Q281 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 1The sum of the infinite geometric series 180 - 60 + 20 - \dots is
  92. 3(b)(i)8 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Let f(x) = e^(-x^2). By calculating the first three non-zero terms and assuming the pattern continues, show that the Maclaurin series expansion of f(x) may be expressed as sum_(k=0)^infinity ((-1)^k x^(2k)) / k!.
  93. 3(b)(ii)3 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Hence, or otherwise, determine the values of x for which the expansion is valid.
  94. 3(c)4 marks· CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2Determine the sum of the series sum_(n=1)^infinity (sin(1/n) - sin(1/(n+1))).
  95. Q191 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The Maclaurin series for \sin x, up to the term in x^3, is
  96. Q201 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1\sum_{r=1}^n \left[\frac{1}{r} - \frac{1}{r+1}\right] =
  97. Q211 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Which of the following series are arithmetic series? \nI. \sum_{r=1}^n (7 + 3r) \nII. \sum_{r=1}^n 2(3^r) \nIII. \sum_{r=1}^n \log_{10} 3^{(r+1)}
  98. Q231 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The sum of the infinite geometric series 180 - 60 + 20 - \dots is
  99. Q241 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Given that S_n = \sum_{i=1}^n \left[\frac{1}{i} - \frac{1}{i+1}\right], \lim_{n\to\infty} S_n is
  100. Q261 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1The value of \sum_{r=1}^\infty 2\left[\frac{1}{4}\right]^{r-1} is
  101. Q271 mark · multiple choice· CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1Given that \sum_{k=1}^n k(k + 1) = S_n, then, for m < n, \sum_{k=m+1}^n k(k + 1) =
  102. 3(a)(i)9 marks· Pure Mathematics · Unit 2 Q3 3(a)(i)Use mathematical induction to prove that 3/4 + 5/36 + ... + (2n-1)/(n²(n-1)²) = 1 - 1/n², n ≥ 1.
  103. 3(a)(i)3 marks· Pure Mathematics · Unit 2 Q3 3(a)(i)Determine the THIRD partial sum, S₃, of the series.
  104. 3(a)(ii)7 marks· Pure Mathematics · Unit 2 Q3 3(a)(ii)Show that Σ (from k=1 to n) 8 / (4k² - 1) = 8n / (2n + 1).
  105. 3(a)(ii)4 marks· Pure Mathematics · Unit 2 Q3 3(a)(ii)Hence, or otherwise, calculate S₃₀ - S₁₀.
  106. 3(b)8 marks· Pure Mathematics · Unit 2 Q3 3(b)Show that if the series 1 + (7 / (3x - 5)) + (7 / (3x - 5))² + (7 / (3x - 5))³ + ... converges, then its sum to infinity is (3x - 8) / (3x - 12).
  107. 3(b)9 marks· Pure Mathematics · Unit 2 Q3 3(b)Determine the Taylor series expansion of f(x) = 1/(1-2x) about x = 3 up to and including the term in x².
  108. 3(b)7 marks· Pure Mathematics · Unit 2 Q3 3(b)Calculate the sum to infinity of the series Σ_{r=2}^{∞} 10/(r² - 1).
  109. 3(b)(i)2 marks· Pure Mathematics · Unit 2 Q3 3(b)(i)Express the series 4 + 4^2 + 4^3 + ... + 4^n using summation notation.
  110. 3(b)(i)8 marks· Pure Mathematics · Unit 2 Q3 3(b)(i)Determine the Taylor series expansion of e^(cos x) about x = π/2 up to the term in x³.
  111. 3(b)(ii)8 marks· Pure Mathematics · Unit 2 Q3 3(b)(ii)Prove by mathematical induction that 4 + 4^2 + 4^3 + ... + 4^n = 4/3 (4^n - 1) for all positive integers n.
  112. 3(b)(ii)3 marks· Pure Mathematics · Unit 2 Q3 3(b)(ii)Use the series expansion to approximate e^(cos π) correct to 2 decimal places.
  113. 3(c)5 marks· Pure Mathematics · Unit 2 Q3 3(c)Determine the Taylor series expansion of x sin(x/2) about x = π, up to and including the first THREE non-zero terms.
  114. 3(c)(ii)5 marks· Pure Mathematics · Unit 2 Q3 3(c)(ii)Hence, use the method of differences to show that Σ(from r=1 to n) 4/((2r+1)(2r+3)) = 2(1/3 - 1/(2n+3)).
  115. 3(c)(iii)3 marks· Pure Mathematics · Unit 2 Q3 3(c)(iii)Hence, calculate Σ(from r=1 to ∞) 4/((2r+1)(2r+3)).
  116. 4(a)3 marks· Pure Mathematics · Unit 2 Q4 4(a)Obtain the Maclaurin series expansion of f(x) = e^(2x) up to and including the term in x^4.
  117. 4(a)(i)8 marks· Pure Mathematics · Unit 2 Q4 4(a)(i)Obtain the Maclaurin series expansion for g up to the term in x⁴.
  118. 4(a)(ii)2 marks· Pure Mathematics · Unit 2 Q4 4(a)(ii)Hence, estimate g(2).
  119. 4(b)(i)2 marks· Pure Mathematics · Unit 2 Q4 4(b)(i)Express the nᵗʰ partial sum S_n of the series using sigma notation.
  120. 4(b)(ii)1 mark· Pure Mathematics · Unit 2 Q4 4(b)(ii)Hence, calculate S₂₀ - S₁₈.
  121. 4(b)(iii)4 marks· Pure Mathematics · Unit 2 Q4 4(b)(iii)Given that Σ(n=1 to ∞) (1/n²) converges, show that S_n diverges.
  122. 4(c)4 marks· Pure Mathematics · Unit 2 Q4 4(c)Calculate ∑(from r=0 to 27) r.
  123. 4(c)7 marks· Pure Mathematics · Unit 2 Q4 4(c)Determine the Maclaurin expansion of f(x) = (1 + x²) cos x up to and including the third non-zero term.
  124. 4(c)8 marks· Pure Mathematics · Unit 2 Q4 4(c)Use the method of induction to prove that Σ(r=1 to n) r(r-1) = n(n² - 1) / 3.
  125. 4(d)6 marks· Pure Mathematics · Unit 2 Q4 4(d)Use the sum of a series to calculate an equivalent fraction for the repeating decimal 0.2727272...

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