Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 32C / a / Solution

An asymptotic sequence satisfiesThe expansion
$f\sim {\sum }_{n\ge 0}{a}_n{\varphi }_n$ means that for every $N\ge 0$,For $N=0$, division by ${\varphi }_0$ givesFor $n\ge 1$, the definition with $N=n$ saysDivision by ${\varphi }_n$ gives the second coefficient formula. In particular, asymptotic-expansion coefficients are unique.