 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/15/b/solution

Since

a_{0} = f (0) \neq = 0

, write

1/ f (x) = b_{0} + b_{1} x + b_{2} x^{2} + O (x^{3})

and multiply the two power series. Equating coefficients with the identity

f (x) / f (x) = 1

gives

a_{0} b_{0} = 1, a_{0} b_{1} + a_{1} b_{0} = 0, a_{0} b_{2} + a_{1} b_{1} + a_{2} b_{0} = 0.

Solving successively yields

\frac{1}{f ( x )} = \frac{1}{a _{0}} - \frac{a _{1}}{a _{0}^{2}} x + \frac{a _{1}^{2} - a _{0} a _{2}}{a _{0}^{3}} x^{2} + O (x^{3}) .

These are the constant, linear and quadratic terms; particular coefficients may vanish. For a convergent power series defining

f

near zero, continuity and

a_{0} \neq = 0

ensure its reciprocal is also defined in a neighborhood of zero.

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