Direct substitution into standard series

1 It is given that \(y = \sinh \left( x ^ { 2 } \right) + \cosh \left( x ^ { 2 } \right)\).

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. Deduce the value of \(\frac { \mathrm { d } ^ { 4 } \mathrm { y } } { \mathrm { dx } ^ { 4 } }\) when \(x = 0\).
3. Use your answer to part (a) to find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { ydx }\), giving your answer as a rational
   fraction in its lowest terms. fraction in its lowest terms.

2

1. Given that \(\mathrm { f } ( x ) = \sin \left( 2 x + \frac { 1 } { 4 } \pi \right)\), show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \sqrt { 2 } ( \sin 2 x + \cos 2 x )\).
2. Hence find the first four terms of the Maclaurin series for \(\mathrm { f } ( x )\). [You may use appropriate results given in the List of Formulae.]

2 It is given that \(\mathrm { f } ( x ) = \ln \left( 1 + x ^ { 2 } \right)\).

1. Using the standard Maclaurin expansion for \(\ln ( 1 + x )\), write down the first four terms in the expansion of \(\mathrm { f } ( x )\), stating the set of values of \(x\) for which the expansion is valid.
2. Hence find the exact value of $$1 - \frac { 1 } { 2 } \left( \frac { 1 } { 2 } \right) ^ { 2 } + \frac { 1 } { 3 } \left( \frac { 1 } { 2 } \right) ^ { 4 } - \frac { 1 } { 4 } \left( \frac { 1 } { 2 } \right) ^ { 6 } + \ldots .$$

10

1. Write down the first three terms of the Maclaurin series for \(\ln \left( 1 + x ^ { 3 } \right)\).
2. Use these three terms to show that \(\ln ( 1.125 ) \approx \frac { n } { 1536 }\), where \(n\) is an integer to be determined.
3. Charlie uses the same first three terms of the series to approximate \(\ln 9\) and gets an answer of 147, correct to 3 significant figures. However, \(\ln 9 = 2.20\) correct to 3 significant figures. Explain Charlie's error.

1. (a) Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos ^ { 2 } \left( \frac { x } { 3 } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\)

Give each term in simplest form.
(b) Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ (c) Use the integration function on your calculator to evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ Give your answer to 5 decimal places.
(d) Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).

9.

1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
2. Find the range of \(x\) for which the expansion is valid.
3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots$$ [BLANK PAGE]

10

1. Using the Maclaurin series for \(\ln ( 1 + x )\), find the first four terms in the series expansion for \(\ln \left( 1 + 3 x ^ { 2 } \right)\).
2. Find the range of \(x\) for which the expansion is valid.
3. Find the exact value of the series $$\frac { 3 ^ { 1 } } { 2 \times 2 ^ { 2 } } - \frac { 3 ^ { 2 } } { 3 \times 2 ^ { 4 } } + \frac { 3 ^ { 3 } } { 4 \times 2 ^ { 6 } } - \frac { 3 ^ { 4 } } { 5 \times 2 ^ { 8 } } + \ldots .$$