Solution
= Solution
Use $\cosh x=1+x^2/2+x^4/24+x^6/720+O(x^8)$. The <Maclaurin series> $\ln(1+u)=u-u^2/2+u^3/3+O(u^4)$ gives
$$
\begin{aligned}
\ln(\cosh x)
&=\frac{x^2}{2}+\left(\frac1{24}-\frac18\right)x^4
+\left(\frac1{720}-\frac1{48}+\frac1{24}\right)x^6+O(x^8)\\
&=\boxed{\frac{x^2}{2}-\frac{x^4}{12}+\frac{x^6}{45}+O(x^8).}
\end{aligned}
$$
All three nonzero terms are needed; stopping at fourth order would supply only two.