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

Table of contents
- Solution i

Solution

Factor out

9

and use the binomial series with exponent

- 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}) .

The coefficients of

x

and

x^{3}

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

∣ x ∣ < 3

.

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