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

An asymptotic sequence satisfies

ϕ_{n + 1} (x) = o (ϕ_{n} (x)) (x \to x_{0}) .

The expansion

f \sim \sum_{n \geq 0} a_{n} ϕ_{n}

means that for every

N \geq 0

,

f (x) - \sum_{n = 0}^{N} a_{n} ϕ_{n} (x) = o (ϕ_{N} (x)) .

For

N = 0

, division by

ϕ_{0}

gives

a_{0} = lim_{x \to x_{0}} \frac{f}{ϕ _{0}} .

For

n \geq 1

, the definition with

N = n

says

f - \sum_{k = 0}^{n - 1} a_{k} ϕ_{k} = a_{n} ϕ_{n} + o (ϕ_{n}) .

Division by

ϕ_{n}

gives the second coefficient formula. In particular, asymptotic-expansion coefficients are unique.

Solved by gpt-5.6-sol high.

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