Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 32E / b

No. Let $x_0=0$ and, for $0<|x|<1$, definewhileBoth families are asymptotic sequences, and ${\psi }_n\sim {\varphi }_n$ for every $n$.

Take $f(x)=1$. It has the exact expansionSuppose it had an expansion $f\sim {\sum }_{n\ge 0}{b}_n{\psi }_n$. The order-zero condition forces $b_0=1$. At order one we would then require some constant $b_1$ such thatAfter division by $|x|$, the left side is $-|x{|}^{-1/2}-b_1$, which cannot tend to zero. This contradiction proves that no such coefficients exist. Thus termwise equivalent asymptotic scales need not preserve expansions.