 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/31d/b/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Keeping the terms through

n = 3

in part (i) gives

I (ϵ) = \frac{1}{1 + ϵ} - \frac{2 ϵ}{( 1 + ϵ ) ^{2}} + \frac{6 ϵ ^{2}}{( 1 + ϵ ) ^{3}} - \frac{24 ϵ ^{3}}{( 1 + ϵ ) ^{4}} + o (ϵ^{3}) .

Expanding the four rational factors in powers of

ϵ

and collecting terms gives

I (ϵ) = 1 - 3 ϵ + 11 ϵ^{2} - 49 ϵ^{3} + o (ϵ^{3}) .

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