4.06b Method of differences: telescoping series

1

1. Given that $$\frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) } = A + B \left( \frac { 1 } { r } - \frac { 1 } { r + 1 } \right)$$ find the values of \(A\) and \(B\).
2. Hence find the value of $$\sum _ { r = 1 } ^ { 99 } \frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) }$$

1

1. Given that \(\mathrm { f } ( r ) = ( r - 1 ) r ^ { 2 }\), show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r ( 3 r + 1 )$$
2. Use the method of differences to find the value of $$\sum _ { r = 50 } ^ { 99 } r ( 3 r + 1 )$$ (4 marks)

2

1. Given that $$\frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { A } { 2 r - 1 } + \frac { B } { 2 r + 1 }$$ find the values of \(A\) and \(B\).
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$
3. Find the least value of \(n\) for which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }\) differs from 0.5 by less than 0.001 .

1

1. Express \(\frac { 1 } { ( r + 2 ) r ! }\) in the form \(\frac { A } { ( r + 1 ) ! } + \frac { B } { ( r + 2 ) ! }\), where \(A\) and \(B\) are integers.
   [0pt] [3 marks]
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 2 ) r ! }\).
   [0pt] [2 marks]

11

1. Show that, for all positive integers \(r\), $$\frac { 1 } { ( r - 1 ) ! } - \frac { 1 } { r ! } = \frac { r - 1 } { r ! }$$ ⟶
   11
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { r - 1 } { r ! } = a + \frac { b } { n ! }$$ where \(a\) and \(b\) are integers to be determined.

9

1. Show that, for \(r > 0\), $$\ln ( r + 2 ) - \ln r = \ln \left( 1 + \frac { 2 } { r } \right)$$ 9
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \ln \left( 1 + \frac { 2 } { r } \right) = \ln \left( \frac { 1 } { 2 } ( n + a ) ( n + b ) \right)$$ where \(a\) and \(b\) are integers to be found.

9

1. Show that, for all positive integers \(r\), $$\frac { r + 1 } { r + 2 } - \frac { r } { r + 1 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$ 9
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) } = \frac { n } { a n + b }$$ where \(a\) and \(b\) are integers to be determined.
   9
3. Hence find the exact value of $$\sum _ { r = 1001 } ^ { 2000 } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$ \(\_\_\_\_\) The curve \(C\) has equation $$y = \frac { 2 x - 10 } { 3 x - 5 }$$ Figure 1 shows the curve \(C\) with its asymptotes. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-12_979_1079_641_468}
   \end{figure}

14

1. Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express \(\sinh ( m + 1 ) x\) and \(\sinh ( m - 1 ) x\) in terms of \(\sinh m x , \cosh m x , \sinh x\) and \(\cosh x\) 14
2. Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of \(\sinh x , \sinh n x\) and \(\sinh ( n + 1 ) x\) Do not write \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}

4

1. Show that $$( r + 1 ) ^ { 2 } - r ^ { 2 } = 2 r + 1$$ 4
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } ( 2 r + 1 ) = n ^ { 2 } + 2 n$$ 4
3. Verify that using the formula for \(\sum _ { r = 1 } ^ { n } r\) gives the same result as that given in part (b).
   [0pt] [3 marks]

1. The complementary function for the second order recurrence relation

$$u _ { n + 2 } + \alpha u _ { n + 1 } + \beta u _ { n } = 20 ( - 3 ) ^ { n } \quad n \geqslant 0$$ is given by $$u _ { n } = A ( 2 ) ^ { n } + B ( - 1 ) ^ { n }$$ where \(A\) and \(B\) are arbitrary non-zero constants.

1. Find the value of \(\alpha\) and the value of \(\beta\). Given that \(2 u _ { 0 } = u _ { 1 }\) and \(u _ { 4 } = 164\)
2. find the solution of this second order recurrence relation to obtain an expression for \(u _ { n }\) in terms of \(n\).
   (6)

2. The general solution of the second order recurrence relation $$u _ { n + 2 } + k _ { 1 } u _ { n + 1 } + k _ { 2 } u _ { n } = 0 \quad n \geqslant 0$$ is given by $$u _ { n } = ( A + B n ) ( - 3 ) ^ { n }$$ where \(A\) and \(B\) are arbitrary non-zero constants.

1. Find the value of \(k _ { 1 }\) and the value of \(k _ { 2 }\) Given that \(u _ { 0 } = u _ { 1 } = 1\)
2. find the value of \(A\) and the value of \(B\).

4

1. Determine an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) giving your answer in the form \(\frac { 1 } { 4 } - \frac { 1 } { 2 } \mathrm { f } ( n )\).
2. Find the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

2 In this question you must show detailed reasoning.

1. Use partial fractions to show that \(\sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } = \frac { 37 } { 60 } - \frac { 1 } { n } - \frac { 1 } { n + 1 } - \frac { 1 } { n + 2 }\).
2. Write down the value of \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).

10

1. You are given that $$\frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { r } - \frac { 2 } { r + 1 } + \frac { 1 } { r + 2 }$$ Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } - \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
2. Hence find the sum of the infinite series $$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$ RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education} \section*{MEI STRUCTURED MATHEMATICS} Further Concepts For Advanced Mathematics (FP1)
   Wednesday 18 JANUARY 2006 Afternoon ..... 1 hour 30 minutes
   Additional materials:
   8 page answer booklet
   Graph paper
   MEI Examination Formulae and Tables (MF2) TIME 1 hour 30 minutes
   • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
   • Answer all the questions.
   • You are permitted to use a graphical calculator in this paper.
   • Final answers should be given to a degree of accuracy appropriate to the context.
   • The number of marks is given in brackets [ ] at the end of each question or part question.
   • You are advised that an answer may receive no marks unless you show sufficient detail of the working to indicate that a correct method is being used.
   • The total number of marks for this paper is 72.

2 Use the method of differences to express \(\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }\) in terms of \(n\), and hence deduce the sum of the infinite series $$\frac { 1 } { 3 } + \frac { 1 } { 15 } + \frac { 1 } { 35 } + \ldots + \frac { 1 } { 4 n ^ { 2 } - 1 } + \ldots$$

13

1. Use the definitions \(\tanh \theta = \frac { \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }\) and \(\operatorname { sech } \theta = \frac { 2 } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }\) to prove the results
   1. \(\tanh ^ { 2 } \theta \equiv 1 - \operatorname { sech } ^ { 2 } \theta\),
   2. \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tanh \theta ) = \operatorname { sech } ^ { 2 } \theta\).
   3. Let \(I _ { n } = \int _ { 0 } ^ { \alpha } \tanh ^ { 2 n } \theta \mathrm {~d} \theta\) for \(n \geqslant 0\), where \(\alpha > 0\).
      (a) Show that \(I _ { n - 1 } - I _ { n } = \frac { \tanh ^ { 2 n - 1 } \alpha } { 2 n - 1 }\) for \(n \geqslant 1\). Given that \(\alpha = \frac { 1 } { 2 } \ln 3\),
      (b) evaluate \(I _ { 0 }\),
   4. use the method of differences to show that \(I _ { n } = \frac { 1 } { 2 } \ln 3 - \sum _ { r = 1 } ^ { n } \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 r - 1 } } { 2 r - 1 }\) and deduce the sum of the infinite series \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } }\).

10

1. Express \(\frac { 1 } { ( k - 1 ) k ( k + 1 ) }\) in partial fractions.
2. Let \(S _ { n } = \sum _ { k = 3 } ^ { n } \frac { 1 } { ( k - 1 ) k ( k + 1 ) }\) for \(n \geqslant 3\). Use the method of differences to show that $$S _ { n } = \frac { 1 } { 12 } - \frac { 1 } { 2 n ( n + 1 ) }$$ and write down the limit of \(S _ { n }\) as \(n \rightarrow \infty\).
3. Given that \(k\) is a positive integer greater than 1 , explain why \(\frac { 1 } { k ^ { 3 } } < \frac { 1 } { ( k - 1 ) k ( k + 1 ) }\).
4. Show that \(\frac { 27 } { 24 } < \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ^ { 3 } } < \frac { 29 } { 24 }\).

11 A sequence of terms \(x _ { n }\) generated by a recurrence relation is said to be strictly increasing if, for each \(x _ { n } , x _ { n + 1 } > x _ { n }\).

1. Let a recurrence relation be defined by $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 } \quad \text { and } \quad x _ { 0 } = \frac { 1 } { 2 } \quad \text { for } n \geq 0$$ Calculate \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) correct to 3 significant figures where appropriate.
2. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 }$$ show that the sequence is strictly increasing when \(x _ { n } > 2\) or \(x _ { n } < 1\).
3. If \(- 1 < x _ { 0 } < 1\), then the sequence \(x _ { n } ( n \geq 0 )\) converges to a limit. Explain briefly why this limit is 1 .
4. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + k } { m } \text { with } m > 0$$ prove that \(x _ { n }\) is a strictly increasing sequence for all \(x _ { n }\) if \(m ^ { 2 } < 4 k\).

1 The \(n\)th triangular number, \(T _ { n }\), is given by the formula \(T _ { n } = \frac { 1 } { 2 } n ( n + 1 )\).

1. Express \(\frac { 1 } { T _ { n } }\) in terms of partial fractions.
2. Hence, using the method of differences, show that \(\sum _ { n = 1 } ^ { N } \left( \frac { 1 } { T _ { n } } \right) = \frac { 2 N } { N + 1 }\).

2

1. Verify that, for all positive values of \(n\), $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find
2. the sum to \(N\) terms,
3. the sum to infinity.

1. Use the method of differences to find \(\sum_{r=1}^{n} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. [4]

It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).

1. Find the value of \(k\). [2]
2. Hence find \(\sum_{r=1}^{n-1} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\). [2]

1. Use the method of differences to find \(\sum_{r=1}^n \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. [4]

It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).

1. Find the value of \(k\). [2]
2. Hence find \(\sum_{r=7}^{n+5} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\). [2]

Given that $$u_n = \frac{1}{n^2 - n + 1} - \frac{1}{n^2 + n + 1},$$ find \(S_N = \sum_{n=N+1}^{2N} u_n\) in terms of \(N\). [3] Find a number \(M\) such that \(S_N < 10^{-20}\) for all \(N > M\). [3]

Answer only one of the following two alternatives. **EITHER** Show that \(\left(n + \frac{1}{2}\right)^3 - \left(n - \frac{1}{2}\right)^3 \equiv 3n^2 + \frac{1}{4}\). [1] Use this result to prove that \(\sum_{n=1}^N n^2 = \frac{1}{6}N(N + 1)(2N + 1)\). [2] The sums \(S\), \(T\) and \(U\) are defined as follows: \begin{align} S &= 1^2 + 2^2 + 3^2 + 4^2 + \ldots + (2N)^2 + (2N + 1)^2,
T &= 1^2 + 3^2 + 5^2 + 7^2 + \ldots + (2N - 1)^2 + (2N + 1)^2,
U &= 1^2 - 2^2 + 3^2 - 4^2 + \ldots - (2N)^2 + (2N + 1)^2. \end{align} Find and simplify expressions in terms of \(N\) for each of \(S\), \(T\) and \(U\). [5] Hence

1. describe the behaviour of \(\frac{S}{T}\) as \(N \to \infty\), [1]
2. prove that if \(\frac{S}{U}\) is an integer then \(\frac{T}{U}\) is an integer. [3]

**OR** The curves \(C_1\) and \(C_2\) have polar equations $$r = 4\cos\theta \quad \text{and} \quad r = 1 + \cos\theta$$ respectively, where \(-\frac{1}{2}\pi \leqslant \theta \leqslant \frac{1}{2}\pi\).

1. Show that \(C_1\) and \(C_2\) meet at the points \(A\left(\frac{4}{3}, \alpha\right)\) and \(B\left(\frac{4}{3}, -\alpha\right)\), where \(\alpha\) is the acute angle such that \(\cos\alpha = \frac{1}{3}\). [2]
2. In a single diagram, draw sketch graphs of \(C_1\) and \(C_2\). [3]
3. Show that the area of the region bounded by the arcs \(OA\) and \(OB\) of \(C_1\), and the arc \(AB\) of \(C_2\), is $$4\pi - \frac{1}{3}\sqrt{2} - \frac{13}{2}\alpha.$$ [7]