4.06b Method of differences: telescoping series

262 questions

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SPS SPS FM Pure 2022 February Q1
7 marks Moderate -0.3
  1. Express \(\frac{1}{(2r-1)(2r+1)}\) in partial fractions. [3]
  2. Hence find \(\sum_{r=1}^{n}\frac{1}{(2r-1)(2r+1)}\), expressing the result as a single fraction. [4]
SPS SPS FM 2021 November Q9
7 marks Standard +0.3
  1. Show that $$\frac{1}{9r - 4} - \frac{1}{9r + 5} = \frac{9}{(9r - 4)(9r + 5)}$$ [2 marks]
  2. Hence use the method of differences to find $$\sum_{r=1}^{n} \frac{1}{(9r - 4)(9r + 5)}.$$ [5 marks]
SPS SPS FM Pure 2023 November Q4
7 marks Standard +0.8
In this question you must show detailed reasoning.
  1. Given that $$\frac{1}{r(r + 1)(r + 2)} = \frac{A}{r(r + 1)} + \frac{B}{(r + 1)(r + 2)}$$ show that \(A = \frac{1}{2}\) and find the value of \(B\). [3]
  2. Use the method of differences to find $$\sum_{r=10}^{98} \frac{1}{r(r + 1)(r + 2)}$$ giving your answer as a rational number. [4]
SPS SPS FM Pure 2025 February Q9
5 marks Challenging +1.2
In this question, you must show detailed reasoning. Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks]
SPS SPS FM Pure 2026 November Q5
5 marks Standard +0.8
Use the method of differences to prove that for \(n > 2\) $$\sum_{r=2}^{n} \frac{4}{r^2-1} = \frac{(pn+q)(n-1)}{n(n+1)}$$ where \(p\) and \(q\) are constants to be determined. [5]
OCR Further Pure Core 2 2018 September Q6
8 marks Challenging +1.2
  1. By considering \(\sum_{r=1}^n ((r+1)^5 - r^5)\) show that \(\sum_{r=1}^n r^4 = \frac{1}{30}n(n+1)(2n+1)(3n^2+3n-1)\). [6]
  2. Use the formula given in part (i) to find \(50^4 + 51^4 + \ldots + 80^4\). [2]
Pre-U Pre-U 9795/1 2011 June Q3
5 marks Challenging +1.2
  1. Express \(\text{f}(r - 1) - \text{f}(r)\) as a single algebraic fraction, where \(\text{f}(r) = \frac{1}{(2r + 1)^2}\). [1]
  2. Hence, using the method of differences, show that $$\sum_{r=1}^{n} \frac{r}{(4r^2 - 1)^2} = \frac{n(n + 1)}{2(2n + 1)^2}$$ for all positive integers \(n\). [4]
Pre-U Pre-U 9795/1 2013 November Q12
10 marks Challenging +1.3
    1. Use the method of differences to prove that $$\sum_{n=k}^N \frac{1}{n(n+1)} = \frac{1}{k} - \frac{1}{N+1}.$$ [4]
    2. Deduce the value of \(\sum_{n=k}^{\infty} \frac{1}{n(n+1)}\) and show that \(\sum_{n=k}^{\infty} \frac{1}{(n+1)^2} < \frac{1}{k}\). [3]
  2. Let \(S = \sum_{n=1}^{\infty} \frac{1}{n^2}\). Show that \(\frac{205}{144} < S < \frac{241}{144}\). [3]
Pre-U Pre-U 9795/1 2015 June Q13
10 marks Challenging +1.2
  1. By sketching a suitable triangle, show that \(\tan^{-1} a + \tan^{-1} \left(\frac{1}{a}\right) = \frac{1}{4}\pi\), for \(a > 0\). [1]
  2. Given that \(a\) and \(b\) are positive and less than 1, express \(\tan(\tan^{-1} a \pm \tan^{-1} b)\) in terms of \(a\) and \(b\). [2]
  3. By letting \(a = \frac{1}{n-1}\) and \(b = \frac{1}{n+1}\), use the method of differences to prove that $$\sum_{n=1}^{\infty} \tan^{-1} \left(\frac{2}{n^2}\right) = \frac{3}{4}\pi.$$ [7]
Pre-U Pre-U 9795/1 2018 June Q1
5 marks Moderate -0.3
  1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
  2. Using the method of differences, prove that \(\sum_{r=1}^{n} \frac{3}{(3r-1)(3r+2)} = \frac{1}{2} - \frac{1}{3n+2}\). [2]
  3. Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(3r-1)(3r+2)}\). [1]
Pre-U Pre-U 9795 Specimen Q4
6 marks Standard +0.3
Write down the sum $$\sum_{n=1}^{2N} n^3$$ in terms of \(N\), and hence find $$1^3 - 2^3 + 3^3 - 4^3 + \ldots - (2N)^3$$ in terms of \(N\), simplifying your answer. [6]
Pre-U Pre-U 9795 Specimen Q9
9 marks Challenging +1.3
Given that \(w_n = 3^{-n} \cos 2n\theta\) for \(n = 1, 2, 3, \ldots\), use de Moivre's theorem to show that $$1 + w_1 + w_2 + w_3 + \ldots + w_{N-1} = \frac{9 - 3\cos 2\theta + 3^{-N+1} \cos 2(N-1)\theta - 3^{-N+2} \cos 2N\theta}{10 - 6\cos 2\theta}.$$ [7] Hence show that the infinite series $$1 + w_1 + w_2 + w_3 + \ldots$$ is convergent for all values of \(\theta\), and find the sum to infinity. [2]