 Solution

ID: past-exam-of-the-perturbation-methods-and-applied-complex-variables-course-of-the-university-of-oxford/2010/a/2/i/solution

a_{n + 1} (ϵ) = o (a_{n} (ϵ))

, the asymptotic expansion condition is that every fixed truncation has remainder smaller than its last retained term:

f (ϵ) - n = 0 \sum N a_{n} (ϵ) = o (a_{N} (ϵ)) for every fixed N .

This limit is taken as

ϵ \to 0

in the specified direction or sector. The number

N

stays fixed while taking that limit. The infinite series need not converge for any nonzero parameter.

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