Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 8 / Solution

The logarithm must be expanded about two, not about one. FactorApplying the supplied logarithm expansion to the bracket gives $\log 2+(x/2+x^2/4)-\frac{1}{2}(x/2{)}^2+O(x^3)$. Thus, through second order,In fact the cubic coefficient vanishes, since $\log (1+e^x)=x/2+\log [2\cosh (x/2)]$ and the second term is even; the Maclaurin series remainder can be sharpened to $O(x^4)$.