 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/15/b/ii/solution

Since

a_{1} = f^{'} (0) \neq = 0

, substitute the proposed inverse series into the forward one. Comparing coefficients of

f, f^{2}, f^{3}

gives

a_{1} b_{1} = 1, a_{1} b_{2} + a_{2} b_{1}^{2} = 0, a_{1} b_{3} + 2 a_{2} b_{1} b_{2} + a_{3} b_{1}^{3} = 0.

b_{1} = \frac{1}{a _{1}}, b_{2} = - \frac{a _{2}}{a _{1}^{3}}, b_{3} = \frac{2 a _{2}^{2} - a _{1} a _{3}}{a _{1}^{5}} .

The conditions

f (0) = 0

and

a_{1} \neq = 0

ensure that both local series are centred at zero and that the inverse has a linear leading term.

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