 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-4/31d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Near the endpoint

t = 0

\frac{1}{1 + t} \sim \sum_{n = 0}^{\infty} (- 1)^{n} t^{n} .

The Watson lemma integrates this local expansion term by term, and

\int_{0}^{\infty} e^{- x t} t^{n} d t = \frac{Γ ( n + 1 )}{x ^{n + 1}} = \frac{n !}{x ^{n + 1}} .

U (x) \sim n = 0 \sum \infty (- 1)^{n} n! x^{- n - 1}, x \to + \infty,

a_{n} = (- 1)^{n} n! .

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