Infinite series convergence and sum

9

1. Show that \(\frac { 1 } { r } - \frac { 1 } { r + 2 } = \frac { 2 } { r ( r + 2 ) }\).
2. Hence find an expression, in terms of \(n\), for $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 2 \times 4 } + \ldots + \frac { 2 } { n ( n + 2 ) }$$
3. Hence find the value of
   (a) \(\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\),
   (b) \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\).

5

1. Show that $$\frac { r + 1 } { r + 2 } - \frac { r } { r + 1 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$
2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 6 } + \frac { 1 } { 12 } + \frac { 1 } { 20 } + \ldots + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$
3. Hence write down the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( r + 1 ) ( r + 2 ) }\).

5

1. Show that $$\frac { 1 } { r } - \frac { 1 } { r + 1 } = \frac { 1 } { r ( r + 1 ) }$$
2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \ldots + \frac { 1 } { n ( n + 1 ) }$$
3. Hence find the value of \(\sum _ { r = n + 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) }\).

5

1. Show that $$\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 } = \frac { 2 } { 4 r ^ { 2 } - 1 }$$
2. Hence find an expression in terms of \(n\) for $$\frac { 2 } { 3 } + \frac { 2 } { 15 } + \frac { 2 } { 35 } + \ldots + \frac { 2 } { 4 n ^ { 2 } - 1 }$$
3. State the value of
   (a) \(\quad \sum _ { r = 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\),
   (b) \(\quad \sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { 4 r ^ { 2 } - 1 }\).

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 $$\frac { 1 } { 1 \times 2 \times 3 } + \frac { 1 } { 2 \times 3 \times 4 } + \frac { 1 } { 3 \times 4 \times 5 } + \ldots$$

9

1. Show that \(\frac { 1 } { 2 r - 3 } - \frac { 1 } { 2 r + 1 } = \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }\).
2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 2 } ^ { n } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 }$$
3. Show that \(\sum _ { r = 2 } ^ { \infty } \frac { 4 } { 4 r ^ { 2 } - 4 r - 3 } = \frac { 4 } { 3 }\).
4. Use an algebraic method to find the square roots of the complex number \(2 + \mathrm { i } \sqrt { 5 }\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are exact real numbers.
5. Hence find, in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are exact real numbers, the roots of the equation $$z ^ { 4 } - 4 z ^ { 2 } + 9 = 0$$
6. Show, on an Argand diagram, the roots of the equation in part (ii).
7. Given that \(\alpha\) is the root of the equation in part (ii) such that \(0 < \arg \alpha < \frac { 1 } { 2 } \pi\), sketch on the same Argand diagram the locus given by \(| z - \alpha | = | z |\).

7

1. Show that \(\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } \equiv \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\).
3. Find \(\sum _ { r = 2 } ^ { \infty } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\).

8

1. Show that \(\frac { r } { r + 1 } - \frac { r - 1 } { r } \equiv \frac { 1 } { r ( r + 1 ) }\).
2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 2 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \ldots + \frac { 1 } { n ( n + 1 ) }$$
3. Hence find \(\sum _ { r = n + 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) }\).

7

1. Show that \(\frac { 1 } { r - 1 } - \frac { 1 } { r + 1 } \equiv \frac { 2 } { r ^ { 2 } - 1 }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 2 } ^ { n } \frac { 2 } { r ^ { 2 } - 1 }\).
3. Find the value of \(\sum _ { r = 1000 } ^ { \infty } \frac { 2 } { r ^ { 2 } - 1 }\).

8

1. Show that \(\frac { 1 } { r } - \frac { 1 } { r + 2 } \equiv \frac { 2 } { r ( r + 2 ) }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 1 } ^ { n } \frac { 2 } { r ( r + 2 ) }\).
3. Given that \(\sum _ { r = N + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) } = \frac { 11 } { 30 }\), find the value of \(N\).

6

1. Show that \(\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 2 ) ^ { 2 } } \equiv \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 1 } ^ { n } \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\).
3. Find \(\sum _ { r = 5 } ^ { \infty } \frac { 4 ( r + 1 ) } { r ^ { 2 } ( r + 2 ) ^ { 2 } }\), giving your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.

8

1. Show that \(\frac { 3 } { r - 1 } - \frac { 2 } { r } - \frac { 1 } { r + 1 } \equiv \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
2. Hence find an expression, in terms of \(n\), for \(\sum _ { r = 2 } ^ { n } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).
3. Hence find the value of \(\sum _ { r = 4 } ^ { \infty } \frac { 4 r + 2 } { r \left( r ^ { 2 } - 1 \right) }\).

9

1. Verify that \(\frac { 4 + r } { r ( r + 1 ) ( r + 2 ) } = \frac { 2 } { r } - \frac { 3 } { r + 1 } + \frac { 1 } { r + 2 }\).
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) } = \frac { 3 } { 2 } - \frac { 2 } { n + 1 } + \frac { 1 } { n + 2 } .$$
3. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) }\) converges as \(n\) tends to infinity.
4. Find \(\sum _ { r = 50 } ^ { 100 } \frac { 4 + r } { r ( r + 1 ) ( r + 2 ) }\), giving your answer to 3 significant figures.

5 You are given that \(\frac { 3 } { ( 5 + 3 x ) ( 2 + 3 x ) } \equiv \frac { 1 } { 2 + 3 x } - \frac { 1 } { 5 + 3 x }\).

1. Use this result to find \(\sum _ { r = 1 } ^ { 100 } \frac { 1 } { ( 5 + 3 r ) ( 2 + 3 r ) }\), giving your answer as an exact fraction.
2. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 + 3 r ) ( 2 + 3 r ) }\) converges as \(n\) tends to infinity.

9 You are given that \(\frac { 3 } { 4 ( 2 r - 1 ) } - \frac { 1 } { 2 r + 1 } + \frac { 1 } { 4 ( 2 r + 3 ) } = \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) }\).

1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 2 } { 3 } - \frac { 3 } { 4 ( 2 n + 1 ) } + \frac { 1 } { 4 ( 2 n + 3 ) } .$$
2. Write down the limit to which \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 5 } { ( 2 r - 1 ) ( 2 r + 1 ) ( 2 r + 3 ) }\) converges as \(n\) tends to infinity.
3. Find the sum of the finite series $$\frac { 45 } { 39 \times 41 \times 43 } + \frac { 47 } { 41 \times 43 \times 45 } + \frac { 49 } { 43 \times 45 \times 47 } + \ldots + \frac { 105 } { 99 \times 101 \times 103 } ,$$ giving your answer to 3 significant figures. \section*{END OF QUESTION PAPER}

3 Given that \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\), show that $$\mathrm { f } ( r - 1 ) - \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$ Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

1 Find the sum of the first \(n\) terms of the series $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$ and deduce the sum to infinity.

2

1. Verify that \(\frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } \left\{ \frac { ( 2 r + 1 ) ( 2 r + 3 ) } { ( r + 1 ) ( r + 2 ) } - \frac { ( 2 r - 1 ) ( 2 r + 1 ) } { r ( r + 1 ) } \right\}\).
2. Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } \left\{ \frac { ( 2 n + 1 ) ( 2 n + 3 ) } { ( n + 1 ) ( n + 2 ) } - \frac { 3 } { 2 } \right\}\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) }\).

The positive variables \(y\) and \(t\) are related by $$y = a ^ { t }$$ where \(a\) is a positive constant.

1. (a) By differentiating \(\ln y\) with respect to \(t\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a\).
   (b) Write down \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Determine the set of values of \(a\) for which the infinite series $$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$ is convergent.
   A curve has parametric equations $$x = t ^ { a } , \quad y = a ^ { t }$$
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(a\) and \(t\), and show that, when \(t = 2\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 Use the method of differences to find \(S _ { N }\), where $$S _ { N } = \sum _ { n = 1 } ^ { N } \frac { 1 } { n ( n + 2 ) }$$ Deduce the value of \(\lim _ { N \rightarrow \infty } S _ { N }\).

1 Verify that \(\frac { 1 } { n ^ { 2 } } - \frac { 1 } { ( n + 1 ) ^ { 2 } } = \frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } }\). Let \(S _ { N } = \sum _ { r = 1 } ^ { N } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\). Express \(S _ { N }\) in terms of \(N\). Let \(S = \lim _ { N \rightarrow \infty } S _ { N }\). Find the least value of \(N\) such that \(S - S _ { N } < 10 ^ { - 16 }\).

1 Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }\).

5

1. Show that \(\lim _ { a \rightarrow \infty } \left( \frac { 3 a + 2 } { 2 a + 3 } \right) = \frac { 3 } { 2 }\).
2. Evaluate \(\int _ { 1 } ^ { \infty } \left( \frac { 3 } { 3 x + 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\), giving your answer in the form \(\ln k\), where \(k\) is a rational number.
   (5 marks)

1 In this question you must show detailed reasoning.

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

8 For each of the following improper integrals, find the value of the integral or explain why it does not have a value:

1. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 3 } { 4 } } \mathrm {~d} x\);
2. \(\int _ { 1 } ^ { \infty } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\);
3. \(\quad \int _ { 1 } ^ { \infty } \left( x ^ { - \frac { 3 } { 4 } } - x ^ { - \frac { 5 } { 4 } } \right) \mathrm { d } x\).