Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 20 / b

Let $u=x^{-5}$, which tends to zero as $|x|\to \infty$. Factor the denominator and use the cosecant expansion near zero:Substitution gives the first two nonzero terms and the requested omitted order:More precisely the next term is $7/(360x^{18})$, followed by $O(x^{-28})$. The outside factor $x^2$ is essential when converting the remainder order from powers of $u$ to powers of $x$.