Is there a quick way to obtain the power series for ln(cos x) ?
Yes there is, though you have to be a bit liberal with some absolute values. $\displaystyle \displaystyle \begin{align*} \int{ -\tan{(x)}\,dx } = \ln{ \left| \cos{(x)} \right| } + C \end{align*}$, so integrate the series for tan(x).
I had done it that way. Is the series valid for the same domain as the tan x series? That is -pi/2 < x < pi/2. I still don't understand the method used at The Student Room.