 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/a/iii/solution

For

F (x) = e^{e^{x}}

, the chain rule gives

F^{'} = e^{x} e^{e^{x}}

and

F^{''} = (e^{x} + e^{2 x}) e^{e^{x}}

. Thus

F (0) = e

F^{'} (0) = e

and

F^{''} (0) = 2 e

, yielding

e^{e^{x}} = e + e x + e x^{2} + O (x^{3}) .

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