4.06b Method of differences: telescoping series

1.(a)By considering the series $$1 + t + t ^ { 2 } + t ^ { 3 } + \ldots + t ^ { n }$$ or otherwise,sum the series $$1 + 2 t + 3 t ^ { 2 } + 4 t ^ { 3 } + \ldots + n t ^ { n - 1 }$$ for \(t \neq 1\) .
(b)Hence find and simplify an expression for $$1 + 2 \times 3 + 3 \times 3 ^ { 2 } + 4 \times 3 ^ { 3 } + \ldots + 2001 \times 3 ^ { 2000 }$$ (c)Write down an expression for both the sums of the series in part(a)for the case where \(t = 1\) .

1. (a) Express \(\frac { 4 r + 2 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions.
   (b) Hence, using the method of differences, prove that

$$\sum _ { r = 1 } ^ { n } \frac { 4 r + 2 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( a n + b ) } { 2 ( n + 1 ) ( n + 2 ) }$$ where \(a\) and \(b\) are constants to be found.

9

1. A clock is designed to chime once each hour, on the hour. The clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 10 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day. Find the probability that the first time it does not chime is
   1. at 0600 on that day,
   2. before 0600 on that day.
   3. Another clock is designed to chime twice each hour: on the hour and at 30 minutes past the hour. This clock has a fault so that each time it is supposed to chime there is a constant probability of \(\frac { 1 } { 20 }\) that it will not chime. It may be assumed that the clock never stops and that faults occur independently. The clock is started at 5 minutes past midnight on a certain day.
      (a) Find the probability that the first time it does not chime is at either 0030 or 0130 on that day.
      (b) Use the formula for the sum to infinity of a geometric progression to find the probability that the first time it does not chime is at 30 minutes past some hour.

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. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \left\{ ( r + 1 ) ^ { 4 } - r ^ { 4 } \right\} = ( n + 1 ) ^ { 4 } - 1$$
2. Show that \(( r + 1 ) ^ { 4 } - r ^ { 4 } \equiv 4 r ^ { 3 } + 6 r ^ { 2 } + 4 r + 1\).
3. Hence show that $$4 \sum _ { r = 1 } ^ { n } r ^ { 3 } = n ^ { 2 } ( n + 1 ) ^ { 2 }$$

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.

5

1. Show that \(\frac { 1 } { 5 r - 2 } - \frac { 1 } { 5 r + 3 } \equiv \frac { 5 } { ( 5 r - 2 ) ( 5 r + 3 ) }\) for all integers \(r\).
2. Hence use the method of differences to show that \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) } = \frac { n } { 3 ( 5 n + 3 ) }\).

5 Use the result \(\frac { 1 } { 5 r - 1 } - \frac { 1 } { 5 r + 4 } \equiv \frac { 5 } { ( 5 r - 1 ) ( 5 r + 4 ) }\) and the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 1 ) ( 5 r + 4 ) }$$ simplifying your answer.

5 Given that \(\frac { 3 } { ( 3 r - 1 ) ( 3 r + 2 ) } \equiv \frac { 1 } { 3 r - 1 } - \frac { 1 } { 3 r + 2 }\), find \(\sum _ { r = 1 } ^ { 20 } \frac { 1 } { ( 3 r - 1 ) ( 3 r + 2 ) }\), giving your answer as an exact fraction.

5

1. Show that \(\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\).
2. Use the method of differences to find \(\sum _ { r = 1 } ^ { 30 } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), expressing your answer as a fraction.

5 You are given that \(\frac { 4 } { ( 4 n - 3 ) ( 4 n + 1 ) } \equiv \frac { 1 } { 4 n - 3 } - \frac { 1 } { 4 n + 1 }\). Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 4 r - 3 ) ( 4 r + 1 ) } = \frac { n } { 4 n + 1 }$$

4 Use the identity \(\frac { 1 } { 2 r + 3 } - \frac { 1 } { 2 r + 5 } \equiv \frac { 2 } { ( 2 r + 3 ) ( 2 r + 5 ) }\) and the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 3 ) ( 2 r + 5 ) }\), expressing your answer as a single fraction.

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 The diagram shows the curve \(y = \frac { 1 } { x ^ { 3 } }\) for \(1 \leqslant x \leqslant n\) where \(n\) is an integer. A set of ( \(n - 1\) ) rectangles of unit width is drawn under the curve. \includegraphics[max width=\textwidth, alt={}, center]{736932f1-4007-4a04-a08b-2551db0b136c-2_611_947_1103_557}

1. Write down the sum of the areas of the rectangles.
2. Hence show that \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 3 } } < \frac { 3 } { 2 }\).

2 Given that $$u _ { n } = \ln \left( \frac { 1 + x ^ { n + 1 } } { 1 + x ^ { n } } \right)$$ where \(x > - 1\), find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(N\) and \(x\). Find the sum to infinity of the series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ when

1. \(- 1 < x < 1\),
2. \(x = 1\).

2 Verify that, for all positive values of \(n\), $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) } .$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find

1. the sum to \(N\) terms,
2. the sum to infinity.

By considering \(\sum _ { k = 0 } ^ { n - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { k }\), show that $$\sum _ { k = 0 } ^ { n - 1 } \cos k \theta \sec ^ { k } \theta = \cot \theta \sin n \theta \sec ^ { n } \theta$$ provided \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\). Hence or otherwise show that $$\sum _ { k = 0 } ^ { n - 1 } 2 ^ { k } \cos \left( \frac { 1 } { 3 } k \pi \right) = \frac { 2 ^ { n } } { \sqrt { 3 } } \sin \left( \frac { 1 } { 3 } n \pi \right)$$ Given that \(0 < x < 1\), show that $$\sum _ { k = 0 } ^ { n - 1 } \frac { \cos \left( k \cos ^ { - 1 } x \right) } { x ^ { k } } = \frac { \sin \left( n \cos ^ { - 1 } x \right) } { x ^ { n - 1 } \sqrt { } \left( 1 - x ^ { 2 } \right) }$$

4 The sum \(S _ { N }\) is defined by \(S _ { N } = \sum _ { n = 1 } ^ { N } n ^ { 5 }\). Using the identity $$\left( n + \frac { 1 } { 2 } \right) ^ { 6 } - \left( n - \frac { 1 } { 2 } \right) ^ { 6 } \equiv 6 n ^ { 5 } + 5 n ^ { 3 } + \frac { 3 } { 8 } n$$ find \(S _ { N }\) in terms of \(N\). [You need not simplify your result.] Hence find \(\lim _ { N \rightarrow \infty } N ^ { - \lambda } S _ { N }\), for each of the two cases

1. \(\lambda = 6\),
2. \(\lambda > 6\).