 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/8/a/i/solution

Multiply the Taylor series

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

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

. Counting the constant as the first nonzero term gives

\frac{cos x}{1 - x} = 1 + x + \frac{x ^{2}}{2} + O (x^{3}) .

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