Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 15 / c

Use the Maclaurin series $\sin x=x-x^3/6+O(x^5)$ and $e^u=1+u+u^2/2+u^3/6+u^4/24+O(u^5)$. The terms through fourth degree giveIn particular the cubic coefficient cancels; it has not been omitted accidentally.

The Maclaurin series give $e^{-x^3}=1-x^3+O(x^6)$ and $\cosh x=1+x^2/2+x^4/24+O(x^6)$. Inverting the latter using the geometric series givesMultiplying the two expansions yieldsThe remainder is $O(x^5)$ because the product already generates an $x^5$ term from $(-x^3)(-x^2/2)$.