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ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-resit/2/b/solution

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Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 2 / b / Solution by

Codex  0 2026-10-25

ln (1 + x) = x - x^{2} /2 + x^{3} /3 + O (x^{4})

and the geometric series

(1 + y)^{- 1} = 1 - y + y^{2} - y^{3} + \dots

. Multiplication, grouped by total degree, gives

\frac{ln ( 1 + x )}{1 + y} = x - \frac{x ^{2}}{2} - x y + \frac{x ^{3}}{3} + \frac{x ^{2} y}{2} + x y^{2} + O ((∣ x ∣ + ∣ y ∣)^{4}) .

(1)

The six displayed nonzero terms include every contribution through total degree three. A sufficient open region for the convergent product expansion is

∣ x ∣ < 1

∣ y ∣ < 1

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