8.01d Sequence limits: limit of nth term as n tends to infinity, steady-states

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. \(\quad a _ { 1 } = 0 \quad a _ { k + 1 } = \left( 1 + a _ { k } \right) ^ { 2 }\).

2

1. Verify that $$\frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } } = \frac { 1 } { n \mathrm { e } ^ { n } } - \frac { 1 } { ( n + 1 ) \mathrm { e } ^ { n + 1 } }$$ Let \(S _ { N } = \sum _ { n = 1 } ^ { N } \frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } }\).
2. Express \(S _ { N }\) in terms of \(N\) and e.
   Let \(S = \lim _ { N \rightarrow \infty } S _ { N }\).
3. Find the least value of \(N\) such that \(( N + 1 ) \left( S - S _ { N } \right) < 10 ^ { - 3 }\).

4

1. Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r - 2 ) } = \frac { 1 } { 3 } - \frac { 1 } { 3 ( 3 N + 1 ) }\).
2. Find the limit, as \(N \rightarrow \infty\), of \(\sum _ { r = N + 1 } ^ { N ^ { 2 } } \frac { N } { ( 3 r + 1 ) ( 3 r - 2 ) }\).

1 Given that $$u _ { n } = \mathrm { e } ^ { n x } - \mathrm { e } ^ { ( n + 1 ) x }$$ find \(\sum _ { n = 1 } ^ { N } \| _ { n }\) in terms of \(N\) and \(x\). Hence determine the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity for cases where this exists.

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 \(S - S _ { N } < 10 ^ { - 16 }\).

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)

2

1. The convergent sequence \(\left\{ \mathrm { a } _ { \mathrm { n } } \right\}\) is defined by \(a _ { 0 } = 1\) and \(\mathrm { a } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { a } _ { \mathrm { n } } } + \frac { 4 } { \sqrt { \mathrm { a } _ { \mathrm { n } } } }\) for \(n \geqslant 0\). Calculate the limit of the sequence.
2. The convergent sequence \(\left\{ \mathrm { b } _ { \mathrm { n } } \right\}\) is defined by \(\mathrm { b } _ { 0 } = 1\) and \(\mathrm { b } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { b } _ { \mathrm { n } } } + \frac { \mathrm { k } } { \sqrt { \mathrm { b } _ { \mathrm { n } } } }\) for \(n \geqslant 0\), where \(k\) is a constant. Determine the value of \(k\) for which the limit of the sequence is 9 .

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 \(\alpha\).

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}

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 \(5 t _ { n + 1 } - 4 t _ { n } = 3 n ^ { 2 } + 28 n + 6\), for \(n \geqslant 0\), with \(t _ { 0 } = 7\).

1. Find an expression for \(t _ { n }\) in terms of \(n\). Another sequence is defined by \(\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { n } ^ { \mathrm { m } } }\), for \(n \geqslant 1\), where \(m\) is a constant.
2. In each of the following cases determine \(\lim _ { n \rightarrow \infty } \mathrm {~V} _ { n }\), if it exists, or show that the sequence is divergent.
   1. \(m = 3\)
   2. \(m = 2\)
   3. \(m = 1\)

2 A sequence is defined by the recurrence relation \(4 \mathrm { t } _ { \mathrm { n } + 1 } - \mathrm { t } _ { \mathrm { n } } = 15 \mathrm { n } + 17\) for \(\mathrm { n } \geqslant 1\), with \(t _ { 1 } = 2\).

1. Solve the recurrence relation to find the particular solution for \(\mathrm { t } _ { \mathrm { n } }\). Another sequence is defined by the recurrence relation \(( n + 1 ) u _ { n + 1 } - u _ { n } ^ { 2 } = 2 n - \frac { 1 } { n ^ { 2 } }\) for \(n \geqslant 1\), with \(u _ { 1 } = 2\).
2. 1. Explain why the recurrence relation for \(\mathrm { u } _ { \mathrm { n } }\) cannot be solved using standard techniques for non-homogeneous first order recurrence relations.
   2. Verify that the particular solution to this recurrence relation is given by \(u _ { n } = a n + \frac { b } { n }\) where \(a\) and \(b\) are constants whose values are to be determined. A third sequence is defined by \(\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { u } _ { \mathrm { n } } }\) for \(n \geqslant 1\).
3. Determine \(\lim _ { n \rightarrow \infty } \mathrm { v } _ { \mathrm { n } }\).

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 \(n \rightarrow \infty\).

3 A tree at the bottom of a garden needs to be reduced in height. The tree is known to increase in height by 15 centimetres each year. On the first day of every year, the height is measured and the tree is immediately trimmed by \(3 \%\) of this height. When the tree is measured, before trimming on the first day of year 1 , the height is 6 metres.
Let \(H _ { n }\) be the height of the tree immediately before trimming on the first day of year \(n\).

1. Explain, in the context of the problem, why the height of the tree may be modelled by the recurrence relation $$H _ { n + 1 } = 0.97 H _ { n } + 0.15 , \quad H _ { 1 } = 6 , \quad n \in \mathbb { Z } ^ { + }$$
2. Prove by induction that \(H _ { n } = 0.97 ^ { n - 1 } + 5 , \quad n \geqslant 1\)
3. Explain what will happen to the height of the tree immediately before trimming in the long term.
4. By what fixed percentage should the tree be trimmed each year if the height of the tree immediately before trimming is to be 4 metres in the long term?

5. A sequence \(\left\{ u _ { n } \right\}\), where \(\mathrm { n } \geqslant 0\), satisfies the second order recurrence relation $$u _ { n + 2 } = \frac { 1 } { 2 } \left( u _ { n + 1 } + u _ { n } \right) + 3 \text { where } u _ { 0 } = 15 \quad u _ { 1 } = 20$$

1. By considering the sequence \(\left\{ v _ { n } \right\}\), where \(u _ { n } = v _ { n } + 2 n\) for \(\mathrm { n } \geqslant 0\), determine an expression for \(u _ { n }\) as a function of n .
2. Describe the long-term behaviour of \(u _ { n }\)

5

1. (a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that \(X _ { 0 } = 12\) and \(X _ { 1 } = 1\).
   (b) Show that the sequence \(\left\{ X _ { n } \right\}\) approaches a geometric sequence as \(n\) increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that \(X _ { n }\) represents the profit (in \(\pounds\) ) at the end of year \(n\).
2. (a) Determine the number of years taken for the projected profit to exceed one million pounds.
   (b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
3. (a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
   (b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
   (c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next \(X _ { n }\) in the process.

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 \(S _ { N } > 4.9\).

1. Find the general solution of $$u_n = 8u_{n-1} - 16u_{n-2}, \quad n \geq 2. \quad (*)$$ [4]

A new sequence \(v_n\) is defined by \(v_n = \frac{u_n}{u_{n-1}}\) for \(n \geq 1\).

1. (A) Use (*) to show that \(v_n = 8 - \frac{16}{v_{n-1}}\) for \(n \geq 2\). [2] (B) Deduce that if \(v_n\) tends to a limit then it must be 4. [2]
2. Use your general solution in part (i) to show that \(\lim_{n \to \infty} v_n = 4\). [3]
3. Deduce the value of \(\lim_{n \to \infty} \left(\frac{u_n}{u_{n-2}}\right)\). [1]

The members of the family of the sequences \(\{u_n\}\) satisfy the recurrence relation $$u_{n+1} = 10u_n - u_{n-1} \text{ for } n \geq 1. \quad (*)$$

1. Determine the general solution of (*). [3]
2. The sequences \(\{a_n\}\) and \(\{b_n\}\) are members of this family of sequences, corresponding to the initial terms \(a_0 = 1\), \(a_1 = 5\) and \(b_0 = 0\), \(b_1 = 2\) respectively.
   1. Find the next two terms of each sequence. [1]
   2. Prove that, for all non-negative integers \(n\), \((a_n)^2 - 6(b_n)^2 = 1\). [8]
   3. Determine \(\lim_{n \to \infty} \frac{a_n}{b_n}\). [2]