Sum geometric series with complex terms

5

1. State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\).
2. Given that \(z\) is an \(n\)th root of unity and \(z \neq 1\), deduce that \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = 0\).
3. Given instead that \(z = \frac { 1 } { 3 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to show that $$\sum _ { m = 1 } ^ { \infty } 3 ^ { - m } \cos m \theta = \frac { 3 \cos \theta - 1 } { 10 - 6 \cos \theta }$$

7

1. State the sum of the series \(1 + \mathrm { w } + \mathrm { w } ^ { 2 } + \mathrm { w } ^ { 3 } + \ldots + \mathrm { w } ^ { \mathrm { n } - 1 }\), for \(w \neq 1\).
2. Show that \(( 1 + i \tan \theta ) ^ { k } = \sec ^ { k } \theta ( \cos k \theta + i \sin k \theta )\), where \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
3. By considering \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { \mathrm { k } }\), show that $$\sum _ { k = 0 } ^ { n - 1 } \sec ^ { k } \theta \sin k \theta = \cot \theta \left( 1 - \sec ^ { n } \theta \cos n \theta \right)$$ provided \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
4. Hence find \(\sum _ { k = 0 } ^ { 6 m - 1 } 2 ^ { k } \sin \left( \frac { 1 } { 3 } k \pi \right)\) in terms of \(m\).

8

1. State the sum of the series \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 }\), for \(z \neq 1\).
2. By letting \(z = \cos \theta + i \sin \theta\), where \(\cos \theta \neq 1\), show that $$1 + \cos \theta + \cos 2 \theta + \ldots + \cos ( n - 1 ) \theta = \frac { 1 } { 2 } \left( 1 - \cos n \theta + \frac { \sin n \theta \sin \theta } { 1 - \cos \theta } \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{dffdf588-eb26-4d08-b1a3-a0226f5e7763-15_833_785_214_680} The diagram shows the curve with equation \(\mathrm { y } = \cos \mathrm { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
3. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } \cos x d x < \frac { 1 } { 2 n } \left( 1 - \cos 1 + \frac { \sin 1 \sin \frac { 1 } { n } } { 1 - \cos \frac { 1 } { n } } \right)$$
4. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \cos x d x\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

8

1. State the sum of the series \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 }\), for \(z \neq 1\).
2. By letting \(z = \cos \theta + i \sin \theta\), where \(\cos \theta \neq 1\), show that $$1 + \cos \theta + \cos 2 \theta + \ldots + \cos ( n - 1 ) \theta = \frac { 1 } { 2 } \left( 1 - \cos n \theta + \frac { \sin n \theta \sin \theta } { 1 - \cos \theta } \right)$$ \includegraphics[max width=\textwidth, alt={}, center]{8b6fd4fe-5ae3-4364-9e97-c67b7606a41e-15_833_785_214_680} The diagram shows the curve with equation \(\mathrm { y } = \cos \mathrm { x }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
3. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } \cos x d x < \frac { 1 } { 2 n } \left( 1 - \cos 1 + \frac { \sin 1 \sin \frac { 1 } { n } } { 1 - \cos \frac { 1 } { n } } \right)$$
4. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \cos x d x\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The series \(C\) and \(S\) are defined for \(0 < \theta < \pi\) by $$\begin{aligned} & C = 1 + \cos \theta + \cos 2 \theta + \cos 3 \theta + \cos 4 \theta + \cos 5 \theta \\ & S = \quad \sin \theta + \sin 2 \theta + \sin 3 \theta + \sin 4 \theta + \sin 5 \theta \end{aligned}$$

1. Show that \(C + \mathrm { i } S = \frac { \mathrm { e } ^ { 3 \mathrm { i } \theta } - \mathrm { e } ^ { - 3 \mathrm { i } \theta } } { \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { i } \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { i } \theta } } \mathrm { e } ^ { \frac { 5 } { 2 } \mathrm { i } \theta }\).
2. Deduce that \(C = \sin 3 \theta \cos \frac { 5 } { 2 } \theta \operatorname { cosec } \frac { 1 } { 2 } \theta\) and write down the corresponding expression for \(S\).
3. Hence find the values of \(\theta\), in the range \(0 < \theta < \pi\), for which \(C = S\).

2 In this question, \(\theta\) is a real number with \(0 < \theta < \frac { 1 } { 6 } \pi\), and \(w = \frac { 1 } { 2 } \mathrm { e } ^ { 3 \mathrm { j } \theta }\).

1. State the modulus and argument of each of the complex numbers $$w , \quad w ^ { * } \quad \text { and } \quad \mathrm { j } w .$$ Illustrate these three complex numbers on an Argand diagram.
2. Show that \(( 1 + w ) \left( 1 + w ^ { * } \right) = \frac { 5 } { 4 } + \cos 3 \theta\). Infinite series \(C\) and \(S\) are defined by $$\begin{aligned} & C = \cos 2 \theta - \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 8 \theta - \frac { 1 } { 8 } \cos 11 \theta + \ldots \\ & S = \sin 2 \theta - \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 8 \theta - \frac { 1 } { 8 } \sin 11 \theta + \ldots \end{aligned}$$
3. Show that \(C = \frac { 4 \cos 2 \theta + 2 \cos \theta } { 5 + 4 \cos 3 \theta }\), and find a similar expression for \(S\).

2

1. You are given the complex numbers \(w = 3 \mathrm { e } ^ { - \frac { 1 } { 12 } \pi \mathrm { j } }\) and \(z = 1 - \sqrt { 3 } \mathrm { j }\).
   1. Find the modulus and argument of each of the complex numbers \(w , z\) and \(\frac { w } { z }\).
   2. Hence write \(\frac { w } { z }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
2. In this part of the question, \(n\) is a positive integer and \(\theta\) is a real number with \(0 < \theta < \frac { \pi } { n }\).
   1. Express \(\mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { j } \theta } + \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta }\) in simplified trigonometric form, and hence, or otherwise, show that $$1 + \mathrm { e } ^ { \mathrm { j } \theta } = 2 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta } \cos \frac { 1 } { 2 } \theta$$ Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \binom { n } { 3 } \cos 3 \theta + \ldots + \binom { n } { n } \cos n \theta \\ & S = \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \binom { n } { 3 } \sin 3 \theta + \ldots + \binom { n } { n } \sin n \theta \end{aligned}$$
   2. Find \(C\) and \(S\), and show that \(\frac { S } { C } = \tan \frac { 1 } { 2 } n \theta\).

2

1. Find the 4th roots of 16j, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate the 4th roots on an Argand diagram.
2. 1. Show that \(\left( 1 - 2 \mathrm { e } ^ { \mathrm { j } \theta } \right) \left( 1 - 2 \mathrm { e } ^ { - \mathrm { j } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 2 \cos \theta + 4 \cos 2 \theta + 8 \cos 3 \theta + \ldots + 2 ^ { n } \cos n \theta \\ & S = 2 \sin \theta + 4 \sin 2 \theta + 8 \sin 3 \theta + \ldots + 2 ^ { n } \sin n \theta \end{aligned}$$
   2. Show that \(C = \frac { 2 \cos \theta - 4 - 2 ^ { n + 1 } \cos ( n + 1 ) \theta + 2 ^ { n + 2 } \cos n \theta } { 5 - 4 \cos \theta }\), and find a similar expression for \(S\).

2

1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{aligned} & C = 1 + a \cos \theta + a ^ { 2 } \cos 2 \theta + \ldots \\ & S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{aligned}$$ where \(a\) is a real number and \(| a | < 1\).
   By considering \(C + \mathrm { j } S\), show that \(C = \frac { 1 - a \cos \theta } { 1 + a ^ { 2 } - 2 a \cos \theta }\) and find a corresponding expression for \(S\).
2. Express the complex number \(z = - 1 + \mathrm { j } \sqrt { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the 4th roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
   Show \(z\) and its 4th roots in an Argand diagram.
   Find the product of the 4th roots and mark this as a point on your Argand diagram.

2

1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{gathered} C = a \cos \theta + a ^ { 2 } \cos 2 \theta + a ^ { 3 } \cos 3 \theta + \ldots \\ S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{gathered}$$ where \(a\) is a real number and \(| a | < 1\).
   By considering \(C + \mathrm { j } S\), show that $$S = \frac { a \sin \theta } { 1 - 2 a \cos \theta + a ^ { 2 } }$$ Find a corresponding expression for \(C\).
2. P is one vertex of a regular hexagon in an Argand diagram. The centre of the hexagon is at the origin. P corresponds to the complex number \(\sqrt { 3 } + \mathrm { j }\).
   1. Find, in the form \(x + \mathrm { j } y\), the complex numbers corresponding to the other vertices of the hexagon.
   2. The six complex numbers corresponding to the vertices of the hexagon are squared to form the vertices of a new figure. Find, in the form \(x + \mathrm { j } y\), the vertices of the new figure. Find the area of the new figure.

7 Let \(S = \mathrm { e } ^ { \mathrm { i } \theta } + \mathrm { e } ^ { 2 \mathrm { i } \theta } + \mathrm { e } ^ { 3 \mathrm { i } \theta } + \ldots + \mathrm { e } ^ { 10 \mathrm { i } \theta }\).

1. (a) Show that, for \(\theta \neq 2 n \pi\), where \(n\) is an integer, $$S = \frac { \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { i } \theta } \left( \mathrm { e } ^ { 10 \mathrm { i } \theta } - 1 \right) } { 2 \mathrm { i } \sin \left( \frac { 1 } { 2 } \theta \right) }$$ (b) State the value of \(S\) for \(\theta = 2 n \pi\), where \(n\) is an integer.
2. Hence show that, for \(\theta \neq 2 n \pi\), where \(n\) is an integer, $$\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos 10 \theta = \frac { \sin \left( \frac { 21 } { 2 } \theta \right) } { 2 \sin \left( \frac { 1 } { 2 } \theta \right) } - \frac { 1 } { 2 }$$
3. Hence show that \(\theta = \frac { 1 } { 11 } \pi\) is a root of \(\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos 10 \theta = 0\) and find another root in the interval \(0 < \theta < \frac { 1 } { 4 } \pi\).

5 Convergent infinite series \(C\) and \(S\) are defined by $$\begin{gathered} C = 1 + \frac { 1 } { 2 } \cos \theta + \frac { 1 } { 4 } \cos 2 \theta + \frac { 1 } { 8 } \cos 3 \theta + \ldots \\ S = \quad \frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots \end{gathered}$$

1. Show that \(C + \mathrm { i } S = \frac { 2 } { 2 - \mathrm { e } ^ { \mathrm { i } \theta } }\).
2. Hence show that \(C = \frac { 4 - 2 \cos \theta } { 5 - 4 \cos \theta }\), and find a similar expression for \(S\).
3. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 17 y = 17 x + 36$$
4. Show that, when \(x\) is large and positive, the solution approximates to a linear function, and state its equation.

10 By considering \(\sum _ { n = 1 } ^ { N } z ^ { 2 n - 1 }\), where \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), show that $$\sum _ { n = 1 } ^ { N } \cos ( 2 n - 1 ) \theta = \frac { \sin ( 2 N \theta ) } { 2 \sin \theta }$$ where \(\sin \theta \neq 0\). Deduce that $$\sum _ { n = 1 } ^ { N } ( 2 n - 1 ) \sin \left[ \frac { ( 2 n - 1 ) \pi } { N } \right] = - N \operatorname { cosec } \frac { \pi } { N }$$

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 $$\sum _ { k = 0 } ^ { n - 1 } \frac { \cos \left( k \cos ^ { - 1 } x \right) } { x ^ { k } } = \frac { \sin \left( n \cos ^ { - 1 } x \right) } { x ^ { n - 1 } \sqrt { } \left( 1 - x ^ { 2 } \right) }$$

5 State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\). By letting \(z = \cos \theta + \mathrm { i } \sin \theta\), show that $$\cos \theta + \cos 2 \theta + \cos 3 \theta + \ldots + \cos n \theta = \frac { \sin \frac { 1 } { 2 } n \theta } { \sin \frac { 1 } { 2 } \theta } \cos \frac { 1 } { 2 } ( n + 1 ) \theta$$ where \(\sin \frac { 1 } { 2 } \theta \neq 0\).

8 By considering \(\sum _ { r = 1 } ^ { n } z ^ { 2 r - 1 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), show that, if \(\sin \theta \neq 0\), $$\sum _ { r = 1 } ^ { n } \sin ( 2 r - 1 ) \theta = \frac { \sin ^ { 2 } n \theta } { \sin \theta }$$ Deduce that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) \cos \left[ \frac { ( 2 r - 1 ) \pi } { 2 n } \right] = - \operatorname { cosec } \left( \frac { \pi } { 2 n } \right) \cot \left( \frac { \pi } { 2 n } \right)$$

6 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Use the binomial expansion of \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, to show that $$\binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \ldots + \binom { n } { n } \cos n \theta = 2 ^ { n } \cos ^ { n } \left( \frac { 1 } { 2 } \theta \right) \cos \left( \frac { 1 } { 2 } n \theta \right) - 1$$ Find $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$

15

1. Show that \(\left( 1 - \frac { 1 } { 4 } \mathrm { e } ^ { 2 \mathrm { i } \theta } \right) \left( 1 - \frac { 1 } { 4 } \mathrm { e } ^ { - 2 \mathrm { i } \theta } \right) = \frac { 1 } { 16 } ( 17 - 8 \cos 2 \theta )\)
   [0pt] [3 marks]
   15
2. Given that the series \(\mathrm { e } ^ { 2 \mathrm { i } \theta } + \frac { 1 } { 4 } \mathrm { e } ^ { 4 \mathrm { i } \theta } + \frac { 1 } { 16 } \mathrm { e } ^ { 6 \mathrm { i } \theta } + \frac { 1 } { 64 } \mathrm { e } ^ { 8 \mathrm { i } \theta } + \ldots\). has a sum to infinity, express this sum to infinity in terms of \(\mathrm { e } ^ { 2 \mathrm { i } \theta }\)
   15
3. Hence show that \(\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 4 ^ { n - 1 } } \cos 2 n \theta = \frac { 16 \cos 2 \theta - 4 } { 17 - 8 \cos 2 \theta }\)
   [0pt] [4 marks]
   15
4. Deduce a similar expression for \(\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 4 ^ { n - 1 } } \sin 2 n \theta\)
   [0pt] [1 mark]

10 Let \(C = \sum _ { r = 0 } ^ { 20 } \binom { 20 } { r } \cos r \theta\). Show that \(C = 2 ^ { 20 } \cos ^ { 20 } \left( \frac { 1 } { 2 } \theta \right) \cos 10 \theta\).

16

1. Show that \(\left( 2 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 2 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by
   \(C = \frac { 1 } { 2 } \cos \theta + \frac { 1 } { 4 } \cos 2 \theta + \frac { 1 } { 8 } \cos 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \cos n \theta\),
   \(S = \frac { 1 } { 2 } \sin \theta + \frac { 1 } { 4 } \sin 2 \theta + \frac { 1 } { 8 } \sin 3 \theta + \ldots + \frac { 1 } { 2 ^ { n } } \sin n \theta\).
2. Show that \(C = \frac { 2 ^ { n } ( 2 \cos \theta - 1 ) - 2 \cos ( n + 1 ) \theta + \cos n \theta } { 2 ^ { n } ( 5 - 4 \cos \theta ) }\).

14

1. Find \(\left( 3 - \mathrm { e } ^ { 2 \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - 2 \mathrm { i } \theta } \right)\) in terms of \(\cos 2 \theta\).
2. Hence show that the sum of the infinite series \(\sin \theta + \frac { 1 } { 3 } \sin 3 \theta + \frac { 1 } { 9 } \sin 5 \theta + \frac { 1 } { 27 } \sin 7 \theta + \ldots\) can be expressed as \(\frac { 6 \sin \theta } { 5 - 3 \cos 2 \theta }\).

13 The complex number \(z\) is defined as \(z = \frac { 1 } { 3 } \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < \frac { 1 } { 2 } \pi\).
On an Argand diagram, the point O represents the complex number 0 , and the points \(P _ { 1 } , P _ { 2 } , P _ { 3 } , \ldots\) represent the complex numbers \(z , z ^ { 2 } , z ^ { 3 } , \ldots\) respectively.

1. Write down each of the following.
   1. The ratio of the lengths \(\mathrm { OP } _ { n + 1 } : \mathrm { OP } _ { n }\)
   2. The angle \(\mathrm { P } _ { n + 1 } \mathrm { OP } _ { n }\)
2. 1. Show that \(\left( 3 - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( 3 - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = \mathrm { a } + \mathrm { b } \cos \theta\), where \(a\) and \(b\) are integers to be determined.
   2. By considering the sum to infinity of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots\), show that $$\frac { 1 } { 3 } \sin \theta + \frac { 1 } { 9 } \sin 2 \theta + \frac { 1 } { 27 } \sin 3 \theta + \ldots = \frac { 3 \sin \theta } { 10 - 6 \cos \theta } .$$

1. The infinite series C and S are defined by

$$\begin{aligned} & \mathrm { C } = \cos \theta + \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 9 \theta + \frac { 1 } { 8 } \cos 13 \theta + \ldots \\ & \mathrm { S } = \sin \theta + \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 9 \theta + \frac { 1 } { 8 } \sin 13 \theta + \ldots \end{aligned}$$ Given that the series C and S are both convergent,

1. show that $$\mathrm { C } + \mathrm { iS } = \frac { 2 \mathrm { e } ^ { \mathrm { i } \theta } } { 2 - \mathrm { e } ^ { 4 \mathrm { i } \theta } }$$
2. Hence show that $$\mathrm { S } = \frac { 4 \sin \theta + 2 \sin 3 \theta } { 5 - 4 \cos 4 \theta }$$

9 In this question you must show detailed reasoning.

1. Show that \(\mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta } = 2 \mathrm { i } \sin \theta\).
2. Hence, show that \(\frac { 2 } { \mathrm { e } ^ { 2 \mathrm { i } \theta } - 1 } = - ( 1 + \mathrm { i } \cot \theta )\).
3. Two series, \(C\) and \(S\), are defined as follows. $$\begin{aligned} & C = 2 + 2 \cos \frac { \pi } { 10 } + 2 \cos \frac { \pi } { 5 } + 2 \cos \frac { 3 \pi } { 10 } + 2 \cos \frac { 2 \pi } { 5 } \\ & S = 2 \sin \frac { \pi } { 10 } + 2 \sin \frac { \pi } { 5 } + 2 \sin \frac { 3 \pi } { 10 } + 2 \sin \frac { 2 \pi } { 5 } \end{aligned}$$ By considering \(C + \mathrm { i } S\), find a simplified expression for \(C\) in terms of only integers and \(\cot \frac { \pi } { 20 }\).
4. Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so. \section*{END OF QUESTION PAPER}