Sequences and series, recurrence and convergence

Prove that $$\sum_{r=1}^{n} 6(r^2 - 1) = (n - 1)n(2n + 5).$$ [4]

11

1. Evaluate \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).
2. Show that Euler's approximate formula, as given in line 13, gives the exact value of \(\sum _ { r = 1 } ^ { 5 } r ^ { 2 }\).

13 Prove that Euler's approximate formula, as given in line 13, when applied to \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { r } ^ { 2 }\) gives exactly \(\frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 }\).

Prove by induction that, for \(n \in \mathbb{Z}^+\), $$\sum_{r=1}^n \frac{1}{r(r+1)} = \frac{n}{n+1}$$ [5]

1. Show that \(\frac{1}{r+2} - \frac{1}{r+3} = \frac{1}{(r+2)(r+3)}\). [2]
2. Hence use the method of differences to find \(\frac{1}{3 \times 4} + \frac{1}{4 \times 5} + \frac{1}{5 \times 6} + \ldots + \frac{1}{52 \times 53}\). [4]

2

1. Use standard results from the List of formulae (MF19) to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( 1 - \mathrm { r } - \mathrm { r } ^ { 2 } \right)\) in terms of \(n\),
   simplifying your answer. simplifying your answer.
2. Show that $$\frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) } = \frac { r + 1 } { ( r + 1 ) ^ { 2 } + 1 } - \frac { r } { r ^ { 2 } + 1 }$$ and hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) }\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) }\).

1 Show that \(\sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 } = \frac { 1 } { 6 } n ( 2 n + 1 ) ( 7 n + 1 )\).

5. (a) Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that $$\sum _ { r = 1 } ^ { n } ( r + 1 ) ( r + 5 ) = \frac { n } { 6 } ( n + 7 ) ( 2 n + 7 )$$ for all positive integers \(n\).
(b) Hence show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) ( r + 5 ) = \frac { 7 n } { 6 } ( n + 1 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.

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 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.

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
   1. \(\sum _ { r = 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\),
   2. \(\sum _ { r = n + 1 } ^ { \infty } \frac { 2 } { r ( r + 2 ) }\).

1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
2. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
3. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).

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]

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]

Prove by the method of differences that \(\sum_{r=1}^{n} r^2 = \frac{1}{6}n(n + 1)(2n + 1)\), \(n > 1\). [6]

9

1. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 3 } - r ^ { 3 } \right\} = ( n + 1 ) ^ { 3 } - 1$$
2. Show that \(( r + 1 ) ^ { 3 } - r ^ { 3 } \equiv 3 r ^ { 2 } + 3 r + 1\).
3. Use the results in parts (i) and (ii) and the standard result for \(\sum _ { r = 1 } ^ { n } r\) to show that $$3 \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 2 } n ( n + 1 ) ( 2 n + 1 )$$

3

1. By considering \(( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ Let \(S _ { n } = 1 ^ { 2 } + 3 \times 2 ^ { 2 } + 3 ^ { 2 } + 3 \times 4 ^ { 2 } + 5 ^ { 2 } + 3 \times 6 ^ { 2 } + \ldots + \left( 2 + ( - 1 ) ^ { n } \right) n ^ { 2 }\).
2. Show that \(\mathrm { S } _ { 2 \mathrm { n } } = \frac { 1 } { 3 } \mathrm { n } ( 2 \mathrm { n } + 1 ) ( \mathrm { an } + \mathrm { b } )\), where \(a\) and \(b\) are integers to be determined.
3. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 3 } }\).

1. Solve the recurrence system

$$\begin{gathered} u _ { 1 } = 1 \quad u _ { 2 } = 4 \\ 9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n \end{gathered}$$

1. A student takes out a loan for \(\pounds 1000\) from a bank.

The bank charges \(0.5 \%\) monthly interest on the amount of the loan yet to be repaid.
At the end of each month

• the interest is added to the loan
• the student then repays \(\pounds 50\)

Let \(U _ { n }\) be the amount of money owed \(n\) months after the loan was taken out.
The amount of money owed by the student is modelled by the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ where \(A\) is a constant.

1. 1. State the value of the constant \(A\).
   2. Explain, in the context of the problem, the value 1.005 Using the value of \(A\) found in part (a)(i),
2. solve the recurrence relation $$U _ { n } = 1.005 U _ { n - 1 } - A \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
3. Hence determine, according to the model, the number of months it will take to completely repay the loan.

12

1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 } .$$
2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
   1. Sketch this curve.
   2. Find the exact length of the curve.

3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$

1. By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
2. Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).

1. A population of deer on a large estate is assumed to increase by \(10 \%\) during each year due to natural causes.

The population is controlled by removing a constant number, \(Q\), of the deer from the estate at the end of each year. At the start of the first year there are 5000 deer on the estate.
Let \(P _ { n }\) be the population of deer at the end of year \(n\).

1. Explain, in the context of the problem, the reason that the deer population is modelled by the recurrence relation $$P _ { n } = 1.1 P _ { n - 1 } - Q , \quad P _ { 0 } = 5000 , \quad n \in \mathbb { Z } ^ { + }$$
2. Prove by induction that \(P _ { n } = ( 1.1 ) ^ { n } ( 5000 - 10 Q ) + 10 Q , \quad n \geqslant 0\)
3. Explain how the long term behaviour of this population varies for different values of \(Q\).

5 In this question, give probabilities correct to 4 decimal places.
Alpha and Delta are two companies which compete for the ownership of insurance bonds. Boyles and Cayleys are companies which trade in these bonds. When a new bond becomes available, it is first acquired by either Boyles or Cayleys. After a certain amount of trading it is eventually owned by either Alpha or Delta. Change of ownership always takes place overnight, so that on any particular day the bond is owned by one of the four companies. The trading process is modelled as a Markov chain with four states, as follows. On the first day, the bond is owned by Boyles or Cayleys, with probabilities \(0.4,0.6\) respectively.
If the bond is owned by Boyles, then on the next day it could be owned by Alpha, Boyles or Cayleys, with probabilities \(0.07,0.8,0.13\) respectively. If the bond is owned by Cayleys, then on the next day it could be owned by Boyles, Cayleys or Delta, with probabilities \(0.15,0.75,0.1\) respectively. If the bond is owned by Alpha or Delta, then no further trading takes place, so on the next day it is owned by the same company.

1. Write down the transition matrix \(\mathbf { P }\).
2. Explain what is meant by an absorbing state of a Markov chain. Identify any absorbing states in this situation.
3. Find the probability that the bond is owned by Boyles on the 10th day.
4. Find the probability that on the 14th day the bond is owned by the same company as on the 10th day.
5. Find the day on which the probability that the bond is owned by Alpha or Delta exceeds 0.9 for the first time.
6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
7. Find the probability that the bond is eventually owned by Alpha. The probabilities that Boyles and Cayleys own the bond on the first day are changed (but all the transition probabilities remain the same as before). The bond is now equally likely to be owned by Alpha or Delta at the end of the trading process.
8. Find the new probabilities for the ownership of the bond on the first day.

Show that $$\sum_{r=11}^{n+1} r^3 = \frac{1}{4}(n^2 + an + b)(n^2 + an + c)$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]

1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { n } { a } ( n + 1 ) ( n + b )$$ where \(a\) and \(b\) are integers to be found.

7. Given that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } ( r + a ) ( r + b ) = \frac { 1 } { 6 } n ( 2 n + 11 ) ( n - 1 )$$ where \(a\) and \(b\) are constants and \(a > b\),

1. find the value of \(a\) and the value of \(b\).
2. Find the value of $$\sum _ { r = 9 } ^ { 20 } ( r + a ) ( r + b )$$

Given \(u_1 = 1\), determine which one of the formulae below defines an increasing sequence for \(n \geq 1\) Circle your answer. [1 mark] \(u_{n+1} = 1 + \frac{1}{u_n}\) \quad \(u_n = 2 - 0.9^{n-1}\) \quad \(u_{n+1} = -1 + 0.5u_n\) \quad \(u_n = 0.9^{n-1}\)

1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by

$$\begin{aligned} a _ { n + 1 } & = 4 - a _ { n } \\ a _ { 1 } & = 3 \end{aligned}$$ Find the value of

1. 1. \(a _ { 2 }\)
   2. \(a _ { 107 }\)
2. \(\sum _ { n = 1 } ^ { 200 } \left( 2 a _ { n } - 1 \right)\)

The integral \(I_n\), where \(n\) is a positive integer, is defined by $$I_n = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^{-n} \sin \pi x \, dx.$$

1. Show that $$n(n+1)I_{n+2} = 2^{n+1} n + \pi - \pi^2 I_n.$$ [5]
2. Find \(I_5\) in terms of \(\pi\) and \(I_1\). [2]

1. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } x ^ { 2 } d x < \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm {~d} x\).

\includegraphics{figure_8} The diagram shows the curve with equation \(y = \frac{1}{x+1}\). A set of \(n\) rectangles of unit width is drawn, starting at \(x = 0\) and ending at \(x = n\), where \(n\) is an integer.

1. By considering the areas of these rectangles, explain why $$\frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n+1} < \ln(n+1).$$ [5]
2. By considering the areas of another set of rectangles, show that $$1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} > \ln(n+1).$$ [2]
3. Hence show that $$\ln(n+1) + \frac{1}{n+1} < \sum_{r=1}^{n+1} \frac{1}{r} < \ln(n+1) + 1.$$ [2]
4. State, with a reason, whether \(\sum_{r=1}^{\infty} \frac{1}{r}\) is convergent. [2]

3 \includegraphics[max width=\textwidth, alt={}, center]{e313d6f0-7615-4be5-b13e-2796fd6335e5-04_540_1511_276_274} The diagram shows the curve \(\mathrm { y } = \frac { \mathrm { x } } { 2 \mathrm { x } ^ { 2 } - 1 }\) for \(x \geqslant 1\), together with a set of \(N - 1\) rectangles of unit
width. width.

1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { r } { 2 r ^ { 2 } - 1 } < \frac { 1 } { 4 } \ln \left( 2 N ^ { 2 } - 1 \right) + 1$$
2. Use a similar method to find, in terms of \(N\), a lower bound for \(\sum _ { r = 1 } ^ { N } \frac { r } { 2 r ^ { 2 } - 1 }\).

The positive integer \(k\) is such that $$\sum_{r=1}^{k} (3r - k) = 90$$ Find the value of \(k\). [3 marks]

1. In this question use the standard results for summations.
   1. Show that for all positive integers \(n\)
   $$\sum _ { r = 1 } ^ { n } \left( 12 r ^ { 2 } + 2 r - 3 \right) = A n ^ { 3 } + B n ^ { 2 }$$ where \(A\) and \(B\) are integers to be determined.
2. Hence determine the value of \(n\) for which $$\sum _ { r = 1 } ^ { 2 n } r ^ { 3 } - \sum _ { r = 1 } ^ { n } \left( 12 r ^ { 2 } + 2 r - 3 \right) = 270$$

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 \(S _ { N }\).

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

1 Let \(\mathrm { f } ( r ) = r ! ( r - 1 )\). Simplify \(\mathrm { f } ( r + 1 ) - \mathrm { f } ( r )\) and hence find \(\sum _ { r = n + 1 } ^ { 2 n } r ! \left( r ^ { 2 } + 1 \right)\).

A sequence is defined by $$u_1 = 600$$ $$u_{n+1} = pu_n + q$$ where \(p\) and \(q\) are constants. It is given that \(u_2 = 500\) and \(u_4 = 356\)

1. Find the two possible values of \(u_3\) [5 marks]
2. When \(u_n\) is a decreasing sequence, the limit of \(u_n\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\). [2 marks]

Write down the sum $$\sum_{n=1}^{2N} n^3$$ in terms of \(N\), and hence find $$1^3 - 2^3 + 3^3 - 4^3 + \ldots - (2N)^3$$ in terms of \(N\), simplifying your answer. [6]

You are given that \(\mathrm{f}(x) = \tan^{-1}(1 + x)\).

1. 1. Find the value of \(\mathrm{f}(0)\). [1]
   2. Determine the value of \(\mathrm{f}'(0)\). [2]
   3. Show that \(\mathrm{f}''(0) = -\frac{1}{2}\). [3]
2. Hence find the Maclaurin series for \(\mathrm{f}(x)\) up to and including the term in \(x^2\). [2]

1 Given that $$u _ { k } = \frac { 1 } { \sqrt { } ( 2 k - 1 ) } - \frac { 1 } { \sqrt { } ( 2 k + 1 ) }$$ express \(\sum _ { k = 13 } ^ { n } u _ { k }\) in terms of \(n\). Deduce the value of \(\sum _ { k = 13 } ^ { \infty } u _ { k }\).

7 Prove that \(\sum _ { r = 1 } ^ { n } \frac { r } { 2 ^ { r - 1 } } = 4 - \frac { n + 2 } { 2 ^ { n - 1 } }\) for all \(n \geqslant 1\).

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]

1. (i) Find the value of

$$\sum _ { r = 4 } ^ { \infty } 20 \times \left( \frac { 1 } { 2 } \right) ^ { r }$$ (3)
(ii) Show that $$\sum _ { n = 1 } ^ { 48 } \log _ { 5 } \left( \frac { n + 2 } { n + 1 } \right) = 2$$

4 In this question you must show detailed reasoning.
Determine the smallest value of \(n\) for which \(\frac { 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } } { 1 + 2 + \ldots + n } > 341\).

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.