Deduce related series from given series

3

1. Find the first three terms of the Maclaurin series for \(\ln ( 2 + x )\).
2. Write down the first three terms of the series for \(\ln ( 2 - x )\), and hence show that, if \(x\) is small, then $$\ln \left( \frac { 2 + x } { 2 - x } \right) \approx x$$

1

1. 1. Use the Maclaurin series for \(\ln ( 1 + x )\) and \(\ln ( 1 - x )\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln \left( \frac { 1 + x } { 1 - x } \right)\). State the range of validity of this series.
   2. Find the value of \(x\) for which \(\frac { 1 + x } { 1 - x } = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places.
2. A curve has polar equation \(r = \frac { a } { 1 + \sin \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
   1. By plotting suitable points, or otherwise, sketch the curve.
   2. Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve.
   3. Obtain the characteristic equation for the matrix \(\mathbf { M }\) where $$\mathbf { M } = \left( \begin{array} { r r r } 3 & 1 & - 2
      0 & - 1 & 0
      2 & 0 & 1 \end{array} \right)$$ Hence or otherwise obtain the value of \(\operatorname { det } ( \mathbf { M } )\).
   4. Show that - 1 is an eigenvalue of \(\mathbf { M }\), and show that the other two eigenvalues are not real. Find an eigenvector corresponding to the eigenvalue - 1 .
      Hence or otherwise write down the solution to the following system of equations. $$\begin{aligned} 3 x + y - 2 z & = - 0.1
      - y & = 0.6
      2 x + z & = 0.1 \end{aligned}$$
   5. State the Cayley-Hamilton theorem and use it to show that $$\mathbf { M } ^ { 3 } = 3 \mathbf { M } ^ { 2 } - 3 \mathbf { M } - 7 \mathbf { I }$$ Obtain an expression for \(\mathbf { M } ^ { - 1 }\) in terms of \(\mathbf { M } ^ { 2 } , \mathbf { M }\) and \(\mathbf { I }\).
   6. Find the numerical values of the elements of \(\mathbf { M } ^ { - 1 }\), showing your working.

6

1. Find and simplify the first five terms in the Maclaurin series for \(\mathrm { e } ^ { 2 x }\)
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2. Hence, or otherwise, write down the first five terms in the Maclaurin series for \(\mathrm { e } ^ { - 2 x }\)
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3. Hence, or otherwise, show that the Maclaurin series for \(\cosh ( 2 x )\) is $$a + b x ^ { 2 } + c x ^ { 4 } + \ldots$$ where \(a\), \(b\) and \(c\) are rational numbers to be determined.

7. (a) Write down the Maclaurin series expansion for \(\ln ( 1 - x )\) as far as the term in \(x ^ { 3 }\).
(b) Show that \(- 2 \ln \left( \frac { 1 - x } { ( 1 + x ) ^ { 2 } } \right)\) can be expressed in the form \(a x + b x ^ { 2 } + c x ^ { 3 } + \ldots\), where \(a , b , c\) are integers whose values are to be determined.

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)