Find series for logarithmic function

2 Find the Maclaurin's series for \(\ln \cosh x\) up to and including the term in \(x ^ { 4 }\).

2

1. Find the coefficient of \(x ^ { 2 }\) in the Maclaurin's series for \(- \ln \cos x\).
2. Find the length of the arc of the curve with equation \(\mathrm { y } = - \operatorname { Incos } \mathrm { x }\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 4 } \pi\).

1 Find the Maclaurin's series for \(\ln \left( 1 + \mathrm { e } ^ { x } \right)\) up to and including the term in \(x ^ { 2 }\).

1 Find the Maclaurin's series for \(\ln ( x + 2 ) + \ln \left( x ^ { 2 } + 5 \right)\) up to and including the term in \(x ^ { 2 }\).

3. $$f ( x ) = \ln ( 1 + \sin k x )$$ where \(k\) is a constant, \(x \in \mathbb { R }\) and \(- \frac { \pi } { 2 } < k x < \frac { 3 \pi } { 2 }\)

1. Find f \({ } ^ { \prime } ( x )\)
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { - k ^ { 2 } } { 1 + \sin k x }\)
3. Find the Maclaurin series of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

14. (a) Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
(b) By considering the root of the equation \(\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0\) deduce that \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).
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