Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 11 - Infinite Series - 11.7 Taylor Series - Exercises - Page 588: 19


$$e^x\sin x =x+x^2+\frac{x^3}{3}+...$$

Work Step by Step

By making use of Table 2, we have the Maclaurin series for $ \sin x$ and $e^x$ as follows $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+...$$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$$ Now, we have the Maclaurin series of $e^x\sin x$ as: $$e^x\sin x=\\ =(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...)(x-\frac{x^3}{3!}+\frac{x^5}{5!}...)\\ =x+x^2+\frac{x^3}{3}+...$$
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