Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2007 / Paper 1 / Section B / 3 / b

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- Solution b

Solution

Since

g (0) = 0

, only the first three positive powers of

g

contribute through degree three. The composition coefficients of power series follow by using

g^{2} = g_{1}^{2} x^{2} + 2 g_{1} g_{2} x^{3} + O (x^{4})

and

g^{3} = g_{1}^{3} x^{3} + O (x^{4})

in

f_{0} + f_{1} g + f_{2} g^{2} + f_{3} g^{3} + \dots

. Hence

f (g (x)) = f_{0} + f_{1} g_{1} x + (f_{1} g_{2} + f_{2} g_{1}^{2}) x^{2} + (f_{1} g_{3} + 2 f_{2} g_{1} g_{2} + f_{3} g_{1}^{3}) x^{3} + O (x^{4}) .

Here

f_{m}

and

g_{m}

are coefficients of the power series, rather than raw derivatives; their definitions already contain any factorial factors.

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