Maclurin Series. (Approximation) Given that $y=\ln \cos x$, show that the first non-zero terms of Maclurin's series for $y=-\frac{x^2}{2}-\frac{x^4}{12}$. Use this series to find the approximation in terms of $\pi$ for $\ln 2$.
My question is how to determine value of $x$ which is suitable?

A suitable value is a value for which $\log\cos x=\log\frac{1}{2}$, for instance $x=\frac{\pi}{3}$.
That gives:
$$ \log 2 \approx \frac{\pi^2}{18}+\frac{\pi^4}{972}.$$
A better choice is $x=\frac{\pi}{4}$, that leads to:
$$ \log 2 \approx \frac{\pi^2}{16}+\frac{\pi^4}{1536}.$$
