 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-2/30a/a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write the remainders after order

N

r_{N} = o ((x - x_{0})^{N})

for

f

and

s_{N} = o ((x - x_{0})^{N})

for

g

. Then

f (x) + g (x) - \sum_{n = 0}^{N} (f_{n} + g_{n}) (x - x_{0})^{n} = r_{N} (x) + s_{N} (x) = o ((x - x_{0})^{N}) .

Hence $h = f + g$ has the termwise sum of the two asymptotic expansions.

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