 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/11/b/i/solution

Factor out

9

- 1/2

(x^{2} + 9)^{- 1/2} = \frac{1}{3} (1 + \frac{x ^{2}}{9})^{- 1/2} = \frac{1}{3} [1 - \frac{1}{2} \frac{x ^{2}}{9} + O (x^{4})] .

Thus the expansion through cubic order is

\frac{1}{x ^{2} + 9} = \frac{1}{3} - \frac{x ^{2}}{54} + O (x^{4}) .

x

and

x^{3}

vanish because the function is an even function. The underlying binomial expansion converges for

∣ x ∣ < 3

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