 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/3/solution

Multiply the standard Taylor series through quadratic order:

e^{x} lo g (1 + x) = (1 + x + \frac{x ^{2}}{2} + \dots) (x - \frac{x ^{2}}{2} + \dots) .

The linear term is

x

and the quadratic coefficient is

1 - 1/2

, hence

f (x) = x + \frac{x ^{2}}{2} + O (x^{3}) .

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