4.08a Maclaurin series: find series for function

1

1. Write down and simplify the first three non-zero terms of the Maclaurin series for \(\ln ( 1 + 3 x )\).
2. Hence find the first three non-zero terms of the Maclaurin series for $$\mathrm { e } ^ { x } \ln ( 1 + 3 x )$$ simplifying the coefficients.

4

1. Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 } + \ldots$$ to write down the first four terms in the expansion, in ascending powers of \(x\), of \(\ln ( 1 - x )\).
2. The function f is defined by $$\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }$$ Use Maclaurin's theorem to show that when \(\mathrm { f } ( x )\) is expanded in ascending powers of \(x\) :
   1. the first three terms are $$1 + x + \frac { 1 } { 2 } x ^ { 2 }$$
   2. the coefficient of \(x ^ { 3 }\) is zero.
3. Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { \sin x } - 1 + \ln ( 1 - x ) } { x ^ { 2 } \sin x }$$ (4 marks)

6 The function f is defined by \(\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\).

1. 1. Find f'''(x).
   2. Using Maclaurin's theorem, show that, for small values of \(x\), $$\mathrm { f } ( x ) \approx 1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
2. Use the expansion of \(\mathrm { e } ^ { x }\) together with the result in part (a)(ii) to show that, for small values of \(x\), $$\mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \approx 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$ where \(k\) is a rational number to be found.
3. Write down the first four terms in the expansion, in ascending powers of \(x\), of \(\mathrm { e } ^ { 2 x }\).
4. Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } - \mathrm { e } ^ { 2 x } } { 1 - \cos x }$$ (4 marks)

6

1. The function f is defined by $$\mathrm { f } ( x ) = \ln \left( 1 + \mathrm { e } ^ { x } \right)$$ Use Maclaurin's theorem to show that when \(\mathrm { f } ( x )\) is expanded in ascending powers of \(x\) :
   1. the first three terms are $$\ln 2 + \frac { 1 } { 2 } x + \frac { 1 } { 8 } x ^ { 2 }$$
   2. the coefficient of \(x ^ { 3 }\) is zero.
2. Hence write down the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right)\).
3. Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \ldots$$ to write down the first three terms in the expansion, in ascending powers of \(x\), of \(\ln \left( 1 - \frac { x } { 2 } \right)\).
4. Use your answers to parts (b) and (c) to find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right) + \ln \left( 1 - \frac { x } { 2 } \right) } { x - \sin x } \right]$$

7 Show that the Maclaurin series for \(\ln ( \mathrm { e } + 2 \mathrm { e } x )\) is $$1 + 2 x - 2 x ^ { 2 } + a x ^ { 3 } - \ldots$$ where \(a\) is to be determined.

15

1. Use Maclaurin's series expansion for \(\ln ( 1 + x )\) to show that the first three terms of the Maclaurin's series expansion of \(\ln ( 1 + 3 x )\) are $$3 x - \frac { 9 } { 2 } x ^ { 2 } + 9 x ^ { 3 }$$ 15
2. Julia attempts to use the series expansion found in part (a) to find an approximation for \(\ln 4\) Julia's incorrect working is shown below. $$\begin{array} { r } \text { Let } 1 + 3 x = 4 \\ 3 x = 3 \\ x = 1 \end{array}$$ $$\text { So } \begin{aligned} \ln 4 & \approx 3 \times 1 - \frac { 9 } { 2 } \times 1 ^ { 2 } + 9 \times 1 ^ { 3 } \\ & \approx 3 - 4.5 + 9 \\ & \approx 7.5 \end{aligned}$$ Explain the error in Julia's working.
   15
3. Use \(x = - \frac { 1 } { 6 }\) in the series expansion found in part (a) to find an approximation for \(\ln 4\) Fully justify your answer.

8

1. The function f is defined as \(\mathrm { f } ( x ) = \sec x\) 8
   1. (i) Show that \(\mathrm { f } ^ { ( 4 ) } ( 0 ) = 5\) 8
   2. (ii) Hence find the first three non-zero terms of the Maclaurin series for \(\mathrm { f } ( x ) = \sec x\) 8
   3. Prove that $$\lim _ { x \rightarrow 0 } \left( \frac { \sec x - \cosh x } { x ^ { 4 } } \right) = \frac { 1 } { 6 }$$

4 In this question you must show detailed reasoning.

1. By writing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\) show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
2. Hence show that \(\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }\).
   1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
   2. By considering the root of the equation \(\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0\) deduce that \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).

5 Let \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )\).

1. 1. Determine \(f ^ { \prime \prime } ( x )\).
   2. Determine the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\).
   3. By considering the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( \mathrm { x } )\), find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer correct to 6 decimal places.
2. By writing \(\mathrm { f } ( x )\) as \(\sin ^ { - 1 } ( x ) \times 1\), determine the value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer in exact form.

1

1. 1. Given that \(\mathrm { f } ( x ) = \arctan x\), write down an expression for \(\mathrm { f } ^ { \prime } ( x )\). Assuming that \(x\) is small, use a binomial expansion to express \(\mathrm { f } ^ { \prime } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 4 }\).
   2. Hence express \(\arctan x\) in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\).
2. Find, in exact form, the value of the following integral. $$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
3. A curve has polar equation \(r = \frac { a } { \sqrt { \theta } }\) where \(a > 0\).
   1. Sketch the curve for \(\frac { \pi } { 4 } \leqslant \theta \leqslant 2 \pi\).
   2. State what happens to \(r\) as \(\theta\) tends to zero.
   3. Find the area of the region enclosed by the part of the curve sketched in part (i) and the lines \(\theta = \frac { \pi } { 4 }\) and \(\theta = 2 \pi\). Give your answer in an exact simplified form.
      1. (i) Express \(2 \sin \frac { 1 } { 2 } \theta \left( \sin \frac { 1 } { 2 } \theta - \mathrm { j } \cos \frac { 1 } { 2 } \theta \right)\) in terms of \(z\) where \(z = \cos \theta + \mathrm { j } \sin \theta\).
         (ii) The series \(C\) and \(S\) are defined as follows. $$\begin{aligned} C & = 1 - \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \cos n \theta \\ S & = - \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \sin n \theta \end{aligned}$$ Show that $$C + \mathrm { j } S = \left\{ - 2 \mathrm { j } \sin \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { j } \sin \frac { 1 } { 2 } \theta \right) \right\} ^ { n } .$$ Hence show that, for even values of \(n\), $$\frac { C } { S } = \cot \left( \frac { 1 } { 2 } n \theta \right)$$
      2. Write the complex number \(z = \sqrt { 6 } + \mathrm { j } \sqrt { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), expressing \(r\) and \(\theta\) as simply as possible. Hence find the cube roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Show the points representing \(z\) and its cube roots on an Argand diagram.
         1. Find the eigenvalues and eigenvectors of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { l l } \frac { 1 } { 2 } & \frac { 1 } { 2 } \\ \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right)$$ Hence express \(\mathbf { M }\) in the form \(\mathbf { P D P } ^ { - 1 }\) where \(\mathbf { D }\) is a diagonal matrix.
         2. Write down an equation for \(\mathbf { M } ^ { n }\) in terms of the matrices \(\mathbf { P }\) and \(\mathbf { D }\). Hence obtain expressions for the elements of \(\mathbf { M } ^ { n }\).
            Show that \(\mathbf { M } ^ { n }\) tends to a limit as \(n\) tends to infinity. Find that limit.
         3. Express \(\mathbf { M } ^ { - 1 }\) in terms of the matrices \(\mathbf { P }\) and \(\mathbf { D }\). Hence determine whether or not \(\left( \mathbf { M } ^ { - 1 } \right) ^ { n }\) tends to a limit as \(n\) tends to infinity. Section B (18 marks)
            1. Given that \(y = \cosh x\), use the definition of \(\cosh x\) in terms of exponential functions to prove that $$x = \pm \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right) .$$
            2. Solve the equation $$\cosh x + \cosh 2 x = 5$$ giving the roots in an exact logarithmic form.
            3. Sketch the curve with equation \(y = \cosh x + \cosh 2 x\). Show on your sketch the line \(y = 5\). Find the area of the finite region bounded by the curve and the line \(y = 5\). Give your answer in an exact form that does not involve hyperbolic functions. \section*{END OF QUESTION PAPER}

6. $$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
2. Determine \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
3. Hence determine the first 3 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)\), giving each coefficient in simplest form.
4. Use the Maclaurin series expansion for \(\ln ( 1 + x )\) to write down the first 4 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln ( 1 + k x )\), where \(k\) is a constant.
5. Hence determine the value of \(k\) for which $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$ exists.

11

1. At all points \(( x , y )\) on the curve \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 0\).
   1. Prove by induction that, for all integers \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n + 1 } y } { \mathrm {~d} x ^ { n + 1 } } + x \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } + n \frac { \mathrm {~d} ^ { n - 1 } y } { \mathrm {~d} x ^ { n - 1 } } = 0$$ where \(\frac { \mathrm { d } ^ { 0 } y } { \mathrm {~d} x ^ { 0 } } = y\).
   2. Given that \(y = 1\) when \(x = 0\), determine the Maclaurin expansion of \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 6 }\).
   3. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = 0\) given that \(y = 1\) when \(x = 0\).
   4. Given that \(Z \sim \mathrm {~N} ( 0,1 )\), use your answers to parts (i) and (ii) to find an approximation, to 4 decimal places, to the probability \(\mathrm { P } ( Z \leqslant 1 )\).
      [0pt] [Note that the probability density function of the standard normal distribution is \(\mathrm { f } ( z ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - \frac { 1 } { 2 } z ^ { 2 } }\).]

8 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).

1. Determine \(f ^ { \prime \prime } ( 1 )\).
2. Differentiate ( \(*\) ) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).

11 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\). \includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-4_631_901_532_571} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed.
2. In the first model the equation is $$y = \mathrm { e } ^ { - x } \cos 12 x$$ Show that this model has a maximum point close to \(A\) and a minimum point close to \(B\), and state the coordinates of these maximum and minimum points and also the \(y\) value when \(x = 0.3\).
3. In an alternative model the equation is $$y = f \cos ( \lambda x ) + g$$ where the constants \(f , \lambda\) and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). Find suitable values for \(f , \lambda\) and \(g\).
4. Using the alternative model, state the value of \(y\) when \(x = 0.3\) and hence comment on how accurate each model is in fitting the three given points.

10

1. Given that \(y = k x \cos x\) is a particular integral for the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 4 \sin x$$ determine the value of \(k\) and find the general solution of this differential equation.
2. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 5 x - 19$$
   1. Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 1\), find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 1\).
   2. Deduce the Taylor series expansion for \(y\) in ascending powers of \(( x - 1 )\), up to and including the term in \(( x - 1 ) ^ { 3 }\), and use this series to find an approximation correct to 3 decimal places for the value of \(y\) when \(x = 1.1\).

11 A curve has polar equation \(r = \mathrm { e } ^ { \sin \theta }\) for \(- \pi < \theta \leqslant \pi\).

1. State the polar coordinates of the point where the curve crosses the initial line.
2. State also the polar coordinates of the points where \(r\) takes its least and greatest values.
3. Sketch the curve.
4. By deriving a suitable Maclaurin series up to and including the term in \(\theta ^ { 2 }\), find an approximation, to 3 decimal places, for the area of the region enclosed by the curve, the initial line and the line \(\theta = 0.3\).

8 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1 ,$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).

1. Determine \(f ^ { \prime \prime } ( 1 )\).
2. Differentiate (*) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).

7 The function \(f\) satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1 ,$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).

1. Determine \(f ^ { \prime \prime } ( 1 )\).
2. Differentiate (*) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).

7 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).

1. Determine \(\mathrm { f } ^ { \prime \prime } ( 1 )\).
2. Differentiate (*) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).

7 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).

1. Determine \(f ^ { \prime \prime } ( 1 )\).
2. Differentiate ( \(*\) ) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
4. Deduce, to 3 decimal places, an approximation for f(1.1).

6

1. Given that \(y = \cos ( \ln ( 1 + x ) )\), prove that
   1. \(\quad ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = - \sin ( \ln ( 1 + x ) )\),
   2. \(( 1 + x ) ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } + y = 0\).
   3. Obtain an equation relating \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   4. Hence find the Maclaurin series for \(y\), up to and including the term in \(x ^ { 3 }\).

1. Starting from the definitions of \(\tanh\) and \(\sech\) in terms of exponentials, prove that $$1 - \tanh^2 \theta = \sech^2 \theta.$$ [3]

The variables \(x\) and \(y\) are such that \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\), for \(-\frac{1}{4}\pi < x < \frac{3}{4}\pi\).

1. By differentiating the equation \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\) with respect to \(x\), show that $$\frac{dy}{dx} = -\operatorname{cosec}\left(x + \frac{1}{4}\pi\right).$$ [4]
2. Hence find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\cos\left(x + \frac{1}{4}\pi\right)\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [5]

It is given that \(y = \sinh(x^2) + \cosh(x^2)\).

1. Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x^4\). [2]
2. Deduce the value of \(\frac{d^4y}{dx^4}\) when \(x = 0\). [1]
3. Use your answer to part (a) to find an approximation to \(\int_0^{\frac{1}{2}} y \, dx\), giving your answer as a rational fraction in its lowest terms. [2]

Find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\frac{1}{2}e^t\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [6]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = e^{1-x}\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).

1. By considering the sum of the areas of these rectangles, show that \(\int_0^1 e^{1-x} \, dx < U_n\), where $$U_n = \frac{e-1}{n(1-e^{-1})}.$$ [4]
2. Use a similar method to find, in terms of \(n\), a lower bound \(L_n\) for \(\int_0^1 e^{1-x} \, dx\). [4]
3. Show that \(\lim_{n \to \infty}(U_n - L_n) = 0\). [2]
4. Use the Maclaurin's series for \(e^x\) given in the list of formulae (MF19) to find the first three terms of the series expansion of \(z(1-e^{-z})\), in ascending powers of \(z\), and deduce the value of \(\lim_{n \to \infty}(U_n)\). [3]