 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/15/c/i/solution

sin x = x - x^{3} /6 + O (x^{5})

and

e^{u} = 1 + u + u^{2} /2 + u^{3} /6 + u^{4} /24 + O (u^{5})

. The terms through fourth degree give

e^{s i n x} = 1 + (x - \frac{x ^{3}}{6}) + \frac{1}{2} (x^{2} - \frac{x ^{4}}{3}) + \frac{x ^{3}}{6} + \frac{x ^{4}}{24} + O (x^{5}) = 1 + x + \frac{x ^{2}}{2} - \frac{x ^{4}}{8} + O (x^{5}) .

In particular the cubic coefficient cancels; it has not been omitted accidentally.

 New to topics? Read the docs here!