Partial fractions then method of differences

4

1. By first expressing \(\frac { 1 } { r ^ { 2 } - 1 }\) in partial fractions, show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } = \frac { 3 } { 4 } - \frac { a n + b } { 2 n ( n + 1 ) }$$ where \(a\) and \(b\) are integers to be found.
2. Deduce the value of \(\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ^ { 2 } - 1 }\).
3. Find the limit, as \(n \rightarrow \infty\), of \(\sum _ { r = n + 1 } ^ { 2 n } \frac { n } { r ^ { 2 } - 1 }\).

2

1. Use standard results from the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 )\) in terms of \(n\),
   fully factorising your answer. fully factorising your answer.
2. Express \(\frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

4 The line \(y = 2 x + 1\) is an asymptote of the curve \(C\) with equation $$y = \frac { x ^ { 2 } + 1 } { a x + b }$$

1. Find the values of the constants \(a\) and \(b\).
2. State the equation of the other asymptote of \(C\).
3. Sketch C. [Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.] \(5 \quad\) Let \(S _ { N } = \sum _ { r = 1 } ^ { N } ( 5 r + 1 ) ( 5 r + 6 )\) and \(T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 5 r + 1 ) ( 5 r + 6 ) }\).

2

1. Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
2. Use the method of differences to find $$\sum _ { r = 1 } ^ { 48 } \frac { 1 } { r ( r + 2 ) }$$ giving your answer as a rational number.

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 $$\frac { 1 } { 3 } + \frac { 1 } { 15 } + \frac { 1 } { 35 } + \ldots + \frac { 1 } { 4 n ^ { 2 } - 1 } + \ldots$$

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 \(\sum _ { n = 1 } ^ { N } \left( \frac { 1 } { T _ { n } } \right) = \frac { 2 N } { N + 1 }\).

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]

Let $$S_N = \sum_{r=1}^{N}(3r + 1)(3r + 4) \quad \text{and} \quad T_N = \sum_{r=1}^{N}\frac{1}{(3r + 1)(3r + 4)}.$$

1. Use standard results from the List of Formulae (MF10) to show that $$S_N = N(3N^2 + 12N + 13).$$ [3]
2. Use the method of differences to show that $$T_N = \frac{1}{12} - \frac{1}{3(3N + 4)}.$$ [3]
3. Deduce that \(\frac{S_N}{T_N}\) is an integer. [2]
4. Find \(\lim_{N \to \infty} \frac{S_N}{N^3 T_N}\). [2]

1. Express \(\frac{2}{(2r + 1)(2r + 3)}\) in partial fractions. [2]
2. Using your answer to (a), find, in terms of \(n\), $$\sum_{r=1}^n \frac{2}{(2r + 1)(2r + 3)}$$ [3]

Give your answer as a single fraction in its simplest form.

Let $$S_n = \sum_{r=1}^{n} \frac{1}{(r+1)(r+3)}$$ where \(n \geq 1\)

1. Use the method of differences to show that $$S_n = \frac{5n^2 + an}{12(n+b)(n+c)}$$ where \(a\), \(b\) and \(c\) are integers. [6 marks]
2. Show that, for any number \(k\) greater than \(\frac{12}{5}\), if the difference between \(\frac{5}{12}\) and \(S_n\) is less than \(\frac{1}{k}\), then $$n > \frac{k-5+\sqrt{k^2+1}}{2}$$ [6 marks]

In this question you must show detailed reasoning.

1. By using partial fractions show that \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2} = \frac{1}{2} - \frac{1}{n+2}\). [5]
2. Hence determine the value of \(\sum_{r=1}^{\infty} \frac{1}{r^2 + 3r + 2}\). [2]

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]

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]