8.01c Sequence behaviour: periodic, convergent, divergent, oscillating, monotonic

7 For each of the following sequences, write down sufficient terms of the sequence in order to be able to describe its behaviour as divergent, periodic or convergent. For any convergent sequence, state its limit.

1. \(a _ { 1 } = - 1 ; \quad a _ { k + 1 } = \frac { 4 } { a _ { k } }\)
2. \(\quad a _ { 1 } = 1 ; \quad a _ { k } = 2 - 2 \times \left( \frac { 1 } { 2 } \right) ^ { k }\)
3. \(\quad a _ { 1 } = 0 \quad a _ { k + 1 } = \left( 1 + a _ { k } \right) ^ { 2 }\).

5 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } ( - 1 ) ^ { r - 1 } r ^ { 2 }\).

1. Use the standard result for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) given in the List of Formulae (MF10) to show that $$S _ { 2 n } = - n ( 2 n + 1 )$$
2. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 2 } }\) and find \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n + 1 } } { n ^ { 2 } }\).

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 The sequence \(\left\{ u _ { n } \right\}\) is defined by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { 12 } { 1 + u _ { n } }\) for \(n \geq 1\).
Given that the sequence converges, with limit \(\alpha\), determine the value of \(\alpha\).

1 The sequence \(\left\{ u _ { n } \right\}\) is defined by \(u _ { 0 } = 2 , u _ { 1 } = 5\) and \(u _ { n } = \frac { 1 + u _ { n - 1 } } { u _ { n - 2 } }\) for \(n \geqslant 2\).
Prove that the sequence is periodic with period 5.

4 The sequence \(\left\{ A _ { n } \right\}\) is given for all integers \(n \geqslant 0\) by \(A _ { n } = \frac { I _ { n + 2 } } { I _ { n } }\), where \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x d x\).

• Show that \(\left\{ A _ { n } \right\}\) increases monotonically.
• Show that \(\left\{ \mathrm { A } _ { \mathrm { n } } \right\}\) converges to a limit, \(A\), whose exact value should be stated.

8 The sequence \(\left\{ u _ { n } \right\}\) of positive real numbers is defined by \(u _ { 1 } = 1\) and \(u _ { n + 1 } = \frac { 2 u _ { n } + 3 } { u _ { n } + 2 }\) for \(n \geqslant 1\).

1. Prove by induction that \(u _ { n } ^ { 2 } - 3 < 0\) for all positive integers \(n\).
2. By considering \(u _ { n + 1 } - u _ { n }\), use the result of part (a) to show that \(u _ { n + 1 } > u _ { n }\) for all positive integers \(n\). The sequence \(\left\{ u _ { n } \right\}\) has a limit for \(n \rightarrow \infty\).
3. Find the limit of the sequence \(\left\{ u _ { n } \right\}\) as \(n \rightarrow \infty\).
4. Describe as fully as possible the behaviour of the sequence \(\left\{ u _ { n } \right\}\). \section*{END OF QUESTION PAPER}

9 For each value of \(k\) the sequence of real numbers \(\left\{ u _ { n } \right\}\) is given by \(u _ { 1 } = 2\) and \(u _ { n + 1 } = \frac { k } { 6 + u _ { n } }\). For each of the following cases, either determine a value of \(k\) or prove that one does not exist.

1. \(\left\{ \mathrm { u } _ { n } \right\}\) is constant.
2. \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is periodic, with period 2 .
3. \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is periodic, with period 4 .

7 In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, \(t\) years after their introduction, is an integer denoted by \(N _ { t }\). The initial number of breeding pairs is given by \(N _ { 0 }\). An initial discrete population model is proposed for \(N _ { t }\). $$\text { Model I: } N _ { t + 1 } = \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right)$$

1. (a) For Model I, show that the steady state values of the number of breeding pairs are 0 and 150 .
   (b) Show that \(N _ { t + 1 } - N _ { t } < 150 - N _ { t }\) when \(N _ { t }\) lies between 0 and 150 .
   (c) Hence find the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when \(N _ { 0 } \in ( 0,150 )\). An alternative discrete population model is proposed for \(N _ { t }\). $$\text { Model II: } N _ { t + 1 } = \operatorname { INT } \left( \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right) \right)$$
2. (a) Given that \(N _ { 0 } = 8\), find the value of \(N _ { 4 }\) for each of the two models.
   (b) Which of the two models gives values for \(N _ { t }\) with the more appropriate level of precision?

1 Three sequences, \(\mathrm { a } _ { \mathrm { n } } , \mathrm { b } _ { \mathrm { n } }\) and \(\mathrm { c } _ { \mathrm { n } }\), are defined for \(n \geqslant 1\) by the following recurrence relations. $$\begin{aligned} & \left( a _ { n + 1 } - 2 \right) \left( 2 - a _ { n } \right) = 3 \text { with } a _ { 1 } = 3 \\ & b _ { n + 1 } = - \frac { 1 } { 2 } b _ { n } + 3 \text { with } b _ { 1 } = 1.5 \\ & c _ { n + 1 } - \frac { c _ { n } ^ { 2 } } { n } = 1 \text { with } c _ { 1 } = 2.5 \end{aligned}$$ The output from a spreadsheet which presents the first 10 terms of \(a _ { n } , b _ { n }\) and \(c _ { n }\), is shown below. Without attempting to solve any recurrence relations, describe the apparent behaviour, including as \(n \rightarrow \infty\), of

• \(a _ { n }\)
• \(\mathrm { b } _ { \mathrm { n } }\)
• \(\mathrm { C } _ { \mathrm { n } }\)

3 A sequence is defined by the recurrence relation \(u _ { n + 2 } = 4 u _ { n + 1 } - 5 u _ { n }\) for \(n \geqslant 0\), with \(u _ { 0 } = 0\) and \(u _ { 1 } = 1\).

1. Find an exact real expression for \(u _ { n }\) in terms of \(n\) and \(\theta\), where \(\tan \theta = \frac { 1 } { 2 }\). A sequence is defined by \(v _ { n } = a ^ { \frac { 1 } { 2 } n } u _ { n }\) for \(n \geqslant 0\), where \(a\) is a positive constant.
2. In each of the following cases, describe the behaviour of \(v _ { n }\) as \(n \rightarrow \infty\).

6 For real constants \(a\) and \(b\), the sequence \(U _ { 1 } , U _ { 2 } , U _ { 3 } , \ldots\) is given by $$U _ { 1 } = a \text { and } U _ { n } = \left( U _ { n - 1 } \right) ^ { 2 } - b \text { for } n \geqslant 2 .$$

1. Determine the behaviour of the sequence in the case where \(a = 1\) and \(b = 3\).
2. In the case where \(b = 6\), find the values of \(a\) for which the sequence is constant.
3. In the case where \(a = - 1\) and \(b = 8\), prove that \(U _ { n }\) is divisible by 7 for all even values of \(n\).

3 In this question you must show detailed reasoning.

1. The sequence \(\left\{ A _ { n } \right\}\) is given by \(A _ { 1 } = \sqrt { 3 }\) and \(A _ { n + 1 } = ( \sqrt { 3 } + 1 ) A _ { n }\) for \(n \geqslant 1\). Find an expression for \(A _ { n }\) in terms of \(n\).
2. The sequence \(\left\{ B _ { n } \right\}\) is given by the formula $$B _ { n } = \frac { 1 } { \sqrt { 3 } } \left( ( \sqrt { 3 } + 1 ) ^ { n } - ( \sqrt { 3 } - 1 ) ^ { n } \right) \text { for } n \geqslant 1 .$$ Explain why \(B _ { n } \rightarrow \frac { 1 } { \sqrt { 3 } } ( \sqrt { 3 } + 1 ) ^ { n }\) as \(n \rightarrow \infty\).
3. The sequence \(\left\{ C _ { n } \right\}\) converges and is defined by \(C _ { n } = \frac { A _ { n } } { B _ { n } }\) for \(n \geqslant 1\). Identify the limit of \(C n\) as \(n \rightarrow \infty\).

7 Irrational numbers can be modelled by sequences \(\left\{ u _ { n } \right\}\) of rational numbers of the form $$u _ { 0 } = 1 \text { and } u _ { n + 1 } = a + \frac { 1 } { b + u _ { n } } \text { for } n \geqslant 0 \text {, }$$ where \(a\) and \(b\) are non-negative integer constants.

1. (a) The constants \(a = 1\) and \(b = 0\) produce the irrational number \(\omega\). State the value of \(\omega\) correct to six decimal places.
   (b) By setting \(u _ { n + 1 }\) and \(u _ { n }\) equal to \(\omega\), determine the exact value of \(\omega\).
2. Use the method of part (i) (b) to find the exact value of the irrational number produced by taking \(a = 0\) and \(b = 2\).
3. Find positive integers \(a\) and \(b\) which would produce the irrational number \(2 + \sqrt { 10 }\). \section*{END OF QUESTION PAPER}

8 A sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by the recurrence system \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = \mathrm { a } - \frac { \mathrm { a } ^ { 2 } } { 2 \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\), where \(a\) is a positive constant.
Determine with justification the behaviour of the sequence for all possible values of \(a\). \section*{END OF QUESTION PAPER}

4 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that, for all positive integers \(n\), $$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$ The sum \(\sum _ { n = 1 } ^ { N } a _ { n }\) is denoted by \(S _ { N }\). Find

1. the value of \(S _ { 30 }\) correct to 3 decimal places,
2. the least value of \(N\) for which \(S _ { N } > 4.9\).

The sequence \(u_1, u_2, u_3, \ldots\) is such that \(u_1 = 1\) and $$u_{n+1} = -1 + \sqrt{(u_n + 7)}.$$

1. Prove by induction that \(u_n < 2\) for all \(n \geqslant 1\). [4]
2. Show that if \(u_n = 2 - \varepsilon\), where \(\varepsilon\) is small, then $$u_{n+1} \approx 2 - \frac{1}{6}\varepsilon.$$ [2]

You are given that \(q \in \mathbb{Z}\) with \(q \geqslant 1\) and that $$S = \frac{1}{(q+1)} + \frac{1}{(q+1)(q+2)} + \frac{1}{(q+1)(q+2)(q+3)} + \cdots$$

1. By considering a suitable geometric series show that \(S < \frac{1}{q}\). [3]
2. Deduce that \(S \notin \mathbb{Z}\). [2]

You are also given that \(\mathrm{e} = \sum_{r=0}^{\infty} \frac{1}{r!}\).

1. Assume that \(\mathrm{e} = \frac{p}{q}\), where \(p\) and \(q\) are positive integers. By writing the infinite series for \(\mathrm{e}\) in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that \(\mathrm{e}\) is irrational. [3]