 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/b/solution

Write

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

and multiply by

a_{0} + a_{1} x + a_{2} x^{2} + O (x^{3})

. The product equals one, so its coefficients give

a_{0} b_{0} = 1

a_{0} b_{1} + a_{1} b_{0} = 0

and

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

. Since

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

\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}) .

This is the initial part of the general reciprocal of a power series recurrence.

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