 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/11/b/i/solution

Multiply the Maclaurin series for the exponential and the sine, keeping terms through degree three:

e^{x} sin x = (1 + x + \frac{x ^{2}}{2} + \frac{x ^{3}}{6} + O (x^{4})) (x - \frac{x ^{3}}{6} + O (x^{5})) .

Only

x

x^{2}

, and

(1/2 - 1/6) x^{3}

contribute at these orders, so

e^{x} sin x = x + x^{2} + \frac{x ^{3}}{3} + O (x^{4}) .

 New to topics? Read the docs here!