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

Sincewe haveThus $a_n=1$ for every $n\ge 0$.

Explicitly,For $n\ge 1$, the remainder after terms $0,\dots ,n-1$ isIts ratio to ${\varphi }_n$ tends to one, verifying every coefficient formula.

Because $e^{-x}=o(x^{-N})$ for every $N$, the exponentially small part is invisible to the power sequence ${\psi }_n=x^{-n}$. Hencewith coefficient one at ${\psi }_1$ and all other power coefficients zero.