Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 11 / b / ii / Solution

Near zero the logarithm's argument is positive, so $\ln [(2+x{)}^3]=3\ln (2+x)=3\ln 2+3\ln (1+x/2)$. Use the Maclaurin series of the natural logarithm:The local calculation uses $x>-2$ for the real logarithm; the power series converges for $|x|<2$.